UNIVERSITY 0lt' NORTH CAROLniA
Department of Statistics
Chapel Hill, N. C.
LINEAR REPRESENMTIONS OF PROJECTIVE PLANES
IN PROJECTIVE SPACES
by
R. H. Bruck
University of Wisconsin and University of North Carolina
and
R. C. Bose
University of North C81"Ol1na
August 1964
•
A linear representation (LR) of a projective plane 7t
(Desarguesian or not) is an isomorphic imbedding of 7t in
~ (Desargues1an) projective apace r: in which each point or
line of 7t becomes a non-empty, finite-dimensional projecti ve subspace ot r: and incidence is given by the containing
relation at E. The paper begins an aXiomatic study of LR's.
At one end of the scale I every finite projective plane With
exactly v lines has a "DOn",dense" LR with E of dimension v.
Towards the other end of the scale, every translation plane
which is suitable "finite-dimensional" has a "dense" LR with
pror:r
close to the lattice-structure of E. A theory
of I . .
-sets" is given, by which new translation
pl.aDee".
blIt derived from old. The paper generalizes an
ear
;ofthe authors (The construction of translation
from 'projective spaces, Journal of Algebra, 1
,.
(1964,'
5-1~2.).
This research was supported by the National Science Foundation Grant
No. GP 16-60 and the Mathematics D:lT1sion of the Air Force Office of
Scientific Research Grant No. 84-63.
Institute of Statistics
Mimeo Series No. 397
,
,
•
LINEAR REPRESENTATIONS OF PROJECTIVE PLANES
IN PROJECTIVE SPACES
by
1
R. H. Bruck and R. C. Bose, University of North Carolina
---------------------------------------------------------~-------------------------
1.
Introduction.
In connection with ~~7 we raised the following question:
Can every (non-Desarguesian) projective plane be imbedded (in som~,!J.atural geometric
fashion) in a (Desarguesian) projective space?
In the present paper, which suggests
one possible version of a theory of linear representations, we feel that we have
made a useful start on an answer.
It should hardly be necessary to point out that
the phrase "natural geometric fashion" admits -- and perhaps deserves -- many
interpretations.
Roughly speaking, the paper splits naturally into two parts.
~
The first part,
comprising sections 2 - 8, is concerned with a general theory of linear representations.
The second part, made up of sections 9 - 14, exploits the particular type
of linear representation given in
[17
to provide a new version of the theory of
translation planes.
Suppose given a (Desarguesian) projective space, E, a "point-set",
" li ne-se t".JC)
, IL f"
e.,
p
and a
If"
"" p' e. t consist of non-empty, finite-dimensional projec t ive
subspaces of E, and if the syatem
(1.1)
consisting of the members of
..
ep
as
"pOints" (point-spaces), the members of etas
"lines" (line-spaces) and the containing relation of E as incidence relation
~ermanent address: Department of Mathematics, University of Wisconsin,
Madison, Wisconsin. The paper was written while Bruck was on leave at the University of North Carolina. The authors had the support of the National Science Foundation Grant No. GP 16-60 and the Air Force Office of Scientific Research Grant
No. 84-63.
2
(point-space contained in line-space), satisfies the aXioms of a projective plane
(our axioms (1), (2), (3),)then we call (1.1) a presentation.
A projective plane n
has a linear representation (1.1) if n is isomorphic to the presentation (1.1).
Every finite projective plane has linear representations in great profusion
(compare Theorems 3.1,
5.4, 5.5 and equations (5.3), (5.4).) The real problem --
which we cannot claim to have solved here -- is to find types of presentations which
will lead to new and deep information about the structure of planes.
In section 4 we introduce four normalization aXioms which are neither automatic
nor (in essence) restrictive for presentations and which are helpful in simplifying
proofs and constructions.
In
section
5 come the lattice axioms (4),(4 1 ) . According to (4), the intersec-
tion of two line-spaces coincides with the unique point-space '!of intersection" ;
and (4') 1s the "point-line dual" of (4).
tions subject to
The only planes with linear representa-
(4) and (4·) are the Desarguesian
finite plane has linear representations subject to
(Theorems
p1~es (T~eorem
5.3), yet every
(4) or (4'), as desired
5.4, 5.5.)
In section 6 we introduce two (dual) concepts of rigidity.
A line-space, R,
of a presentation (1.1) is rigid provided that, for every two point-spaces A, B,
at least one of which is contained in R, the projective SUbspace A + B is the unique
o.
line-space containing A ,.andBo
.
•
space is rigid.)
(in these terms, axiom (4 f) states that every 1ine-
A presentation subject to aXiom (4) and having a rigid line-space,
R, is a translation plane with respect to the "line" R (Theorem 6.1.)
of information about such presentations is given in Lemmas 6.2, 6.3.
A good deal
3
Section 7 contains two completeness axioms (5), (6).
Axiom (5) states that
each point of E is contained in exactly one poinW'pace; axiom (6), that a pointspace is either disjoint from a line-space or entirely contained in the line-space.
In the presence ofaxoim (5), axioms (4) and (6) are equivalent (Lemma 7.1.)
Lemma: 7.2 and Theorem 7.3 give methods of "improving' the properties of a presentation subject to axioms
(4), (5), (6). It might be added here that we do not know
the class of all planes defined by the presentations SUbject to axioms (4), (5), (6).
In section
8 we construct presentations subject to axioms (4), (5), (6) for any
Desarguesian plane over a skew-field F such that F is finite-dimensional over its
center.
Here we may have a2(1an (4 1 ) , or one rigid line-space, or no rigid line-
space, as desired.
In the latter case (see Example 8.3) we encounter a type of
presentation which may perhaps yield planes other than translation planes;
the
possibilities are still qUite unknown.
In section 9 we point out that the construction in
a presentation subject to axioms
LI_7 can be
reinterpreted as
(4), (5), (6), possessing a rigid line-space R
which is a hyperplane of E, and such that the point-spaces not contained in R are
points of E.
To sum up, the first part of the paper presents no new information
about the structure of projective planes but merely lays a foundation for a theory
of their linear representations.
At this point we should like to acknowledge with thanks a letter from Mario
Benedicty to Bruck.
Commenting on ~17, without being aware ot Examples 8.1, 8.2
which we had already obtained -- Benedicty suggested, in essence, the study of
presentations subject to axioms
(4), (5), (6). In so doing, he started us thinking
about the present theory of linear representations.
4
The second part of our paper, devoted to the construction, given in ~17, of
translation planes, is concerned with the theory of spreads.
between three points of view.
We shift at will
Let us first discuss matters here in purely geometric
terms.
By at-spread,,8, over a skew-field F we mean a collection of (t-l) - dimensional projective subspaces over F of a (2t - 1) - dimensional projective space ~.
over F such that each point of
~.
is contained in exactly one member of
spread..2 corresponds a unique translation plane
(for example) when n(~ ) is Desarguesian.
nL8).
~.
To each
We may call ,.8 Desarguesian
A necessary condition that ~ be Pappus
is that the underlying skew-field F be a field.
Now let F be a field.
To each triple A, B, C of mutually disjoint (t - 1) -
dimensional subspaces of ~"t there corresponds a unique regulus 0;.;
namely a collection
a: of
oj
(£(A, B, C),
(t - 1) - dimensional projective sUbspaces of ~' maximal
in the pr0'l{E! r ty that every Une of
a.
Ii;:I
~
t
which meets A, B and C meets every member of
A t - spread ,..8 is called regular if."g contains
distinct members A, B, C of~.
CJ2 (A, B, C) for every three
Every Pappus spread is regular.
two elements, every regular spread is Moufang.
If F has more than
In particular, if F = GF(q), q > 2,
the properties of being regular, Pappus, Desarguesian or Moufang are equivalent for
spreads over F.
All spreads over GF(2) are regular but (as noted more precisely
below) not all are Desarguesian.
Again let ~ t be a (2t - 1) - dimensional projective space over a skew-field F
and let
,.s, ..3
t
be two distinct spreads of ~.t with at least one common member.
~ is the collection of all members of
A
If
which are not members of ,.J -', and if x"is
similarly defined with..J ,.l-t interchanged, then.:t' is an example of a switching set
5
of E l , with )(. as one of its conjugates.
13.)
"S -t,
(For an abstract definition, see section
If a spread Aof E t contains a switching set X , we may derive a new spread,
from
X·.
-<! by replacing X by a conjugate 1
In the case that F = GF (q) we show
that Et always has switching sets; they eXist in greater variety of types the more
prime-factors t has.
As one consequence, there eXist non-Desarguesian t-spreads
over GF(q) (for t >
= 2) except perhaps when q
=2
following is true:
n!s a prime-power
If the positive integer
and t is a prime.
translation plane of order n is Desarguesia~ then n
This result seems to be new for n an odd power of 2.
=p
Therefore the
su,s.~
th8:.t every
or 2Pwh~ p :f.S a prime.
(See Hall
[-!f.7. )
The theory of spreads 1 as sketched above 1 is developed by representing a spread
in terms of linea.r tranB'formations (see section 10.)
For t
sional vector space over a skew-field F of left operators.
linear transformations of W over F such that (i)
ence of each two distinct elements of
on the nonzero elements of W.
To
e
e
>
= 2, let
W be a t-dimen-
e be
a collection of
Let
e contains 0 and I
is nonsingular and (iii)
j
(11) the differ-
C is transitive
corresponds a t-spread over F whose members may
be considered to be the elements of t= together with an additional symbol,
a collection
el.
equivalent to
e
if it can be obtained from
transformations of the follOWing types:
X -> P
(c)
-1
(a)
XP; (b) An inversion X - > X·
1
e
Call
co.
by iteration of
A similarity transformation
(understood to ;interchange 0 and (D);
Andfin-e transformation
X - > (T - S )"-1 (X - S)
(where S, T are distinct, but otherwise arbitrary, elements of
e .)
If.,J,
<!'
J
are equivalent spreads (corresponding to equivalent collections of linear transforma-
6
tiona) then the translation planes
1C
(..J ),
1C (
t-spread over F corresponding to a collection
then the conditiona that
terms of (3.
field.
hand,
...t be
-e:f1 )
are isomorphic.
e of
If..J is a
linear transformations of W,
Moufang, Desarguesian or Pappus can be stated in
For example, -<9 is Desargueaian if and only if
(c:,
+ , .) is a skew-
(See (11.2), (11.3), (11.4), and Theorems 11.1, 11.2, 11.3.)
A!
lent to
On the other
is regular if and only if F is a field and, for each collection
e,
it is true that
e
f
e
f
equiva-
is e"losed under multiplication by elements of F.
(For the implications of regularity, see Theorem 12.1 and Corollary.)
For the
theory of switching sets of PG(2t - 1, q), we regard the finite field GF(qt) as a
collection (corresponding to a Pappus spread) of linear transformation of a t-dimensional vector space over GF(q).
(See Theorems 13.1, 13.2)
Finally, in section 14,
we find 256 spreads of PG(7, 3) by applying the theory of switching sets to a Pappus
spreadj but we do not eJCBmine the equivalence classes of the 256 spreads in detail.:
A third way of looking at spreads may be noted briefly.
If
e
is a collection,
representing a spread, of linear transformations of the t-dimensional vector SP8ce
Wover F, we may turn W into a (right) Veblen-Wedderburn system
ing multiplication as follows:
w(lX)
for every w in W and X in
e.
by defin-
Choose any non-zero element, 1, of W and define
=
wX
If.J is the spread corresponding to
(W, +, .) is a coordinate ring of 7C( . .J
tions of (W, +, .).
(w, +, .)
), and
e is
e , then
the set of all right multiplica-
By shifting our point of view, as convenient, between the three
ways of studying spreads, we develop the theory of spreads without appealing explicitly to the theory of collineations of projective space.
7
In conclusion" we should like to express our thanks to Dale Mesner who pro-
vided a lively audience of one for the evolution of the present paper •
.
-e
8
2.
F.
Preliminaries.
Let E be a (Desarguesian) projective space over a skeW-field
As noted in Bruck and Bose [lJ, we may represent E in the form
E
= E(v/F)
where V is a vector-space (of finite or infinite dimension; depending on E) over F
as a ring of left operators.
Here .. for each non-negative integer s, an (s - 1)-
dimensional projective subs-pace of E is an s-dimensionltl"ve,ctor-subspace of V over F.
(We are using the term "projective subspace" in the narrow sense of "projective subspace over the coordinate skew-field F of E.") In particular, E has finite dimension
d-~precisely when
V has finite dimension dover F.
In the representation (2.1), a point of E is
V over F.
So
l-dimensional vector subspace of
We shall regard the projective subsIlaces of E as po:!ut Gets.
'With this convention.. the zero vector-subspace.. {
0
J..
of V over F is the empty pro-
jective subspace of E (and has projective dimension -1.)
The containing relation of
V induces a containing relation among the projective subspaces of E.
non-empty collection of' projl3ctive subS'paces of E, the intersection,
is the set of all points cotm:lon to all the members of
space of E.
Again.. the ~~ of
e
e..
If
~Y
e
'"' e .
If
is a
of
e
and is a projective sub-
is the intersection of the collection of all
projective subspaces of E which contain. every member of
space of E.
For example..
e,
and is a projective sub-
are projective subspaces of E, X ,A Y and X + Y denote the inter-
section and union, respectively of the (collection of ) subspaces X, Y.
Similarly
for three or more subspaces.
The following simple lemma will prove convenient:
Lemma 2.1.
Let E be a Desarguesian projective 8J)ace end let Eo be a (possibly
empty) projective SUbspace of E.
Then the guotier.;"t space E/E " ~nsisting of all
o
projective subspaces of E which contain Eo"tas'Desarguesian ;p~oject1ve space for
-e
9
which Eo counts as the empty projective subspace.
In addition, there eXists at least
one projective subspace E-' ofc_~~ that E = Eo + E'· ~;g.~ Eo '"
? r 1~ ~~:llz.:.
£;pr
each such E'. the guot:i$nt spe.ce E/E is isomorphic toE~ MOTeover, if Ee-and E 4
o
are :preassi~.!:sarguesianprojective spaces over the same skew-field F, then there
eXists at least on~ Desarguesian F:~jective sp~~!__E over F suc~__~ E/Eo is isomorphic to E '.
Proof.
We may assume, for the first part of the lemma, that E has the repre-
sentation (2.1).
over F.
Then Eo has a unique representa.tion as a vector-subspace Vo of V
The definition of E/E shows at once that
o
where VIv is the familiar quotient vector-space of V over F.
o
be represented in at least one way as a direct sum
V
=
Vo + W
where Wis a vector-subspace of V over F.
E'
=
The vector-space V can
Thus, if
E(w/F),
then E = Eo + E" and E..,.
E' is empty. In addition, E/E0 is isomorphic to Eo'. This
0
proves all but the last sentence of Lemma 2.1; and the last sentence should now be
obvious.
If E is given by (2.1), then the dimension formula
(2.2)
dim A + dim B = dim (A
B) + dim (A + B)
holds if A, B are finite-dimensional vector subspacesof V over F, where dim A denotes
the dimension. of A as a vector space over F.
When we substract 1 from each of the
four terms in (2.2), we see that (2.2) is equally valid if A, B are finite-dimen-
-e
10
sional projective subspaces of E and if dim A denotes the (pro jective) dimension of
A.
In the sequel we shall use (2.2) without comment; it will be clear from the con-
text whether vector-space dimension or projective dimension is intended.
At one or two points in the sequel we shall need the concept of a hyperplane.
A hyperplane of a projective space E is a maximal proper projective subspace of E.
In other words, a projective sUbspace I:
t
of a projective space E is a hyperplane of
B provided (a) there eXists at least one point of E which is not in E' and (b)
.
P + E' = I: for every point P of E which is not in Ei.
3. Linear representations of projective planes. Let us now attempt to represent (not necessarily Desarguesian) projective planes in terms of (Desarguesian)
projective spaces.
e
We want the representations to be linear in the following sense:
E is a (Desarguesian) projective space and
1t
projective plane.
is a non-empty, finite-dimensional projec-
Each point or line of
1t
is a (not necessarily Desarguesian)
tive subspace of 1::, and the incidence relation of
relation of E.
1t
If' we denote the pOint-set of 1t by
is that induced by the containing
ep
and the line-set of
1t
by
el ,
the following aXioms must hold:
(0)
E is a Desarguesian projectiva s1?ace and C?p'
el
are collections of non-
empty, finite-dimensional projective subspaces of E.
(1)
If' A, B are distinct members of
one member of (!!
(2)
member of
(3)
e pl
then A + B is contained in exactly
e {I
then S ro T contains at least one
t
If' S, T are distinct members of
ep.
There exists at least one sat A, B, C, D of four distinct members of <S>p
such that no three of A, B, C. D are contained in the s8lllemember of
When we speak of the presentation
e {-
11
we shall understand that axioms (0) - (3) hold and that n is a system consisting of
the members of
e'P ,
e.
as its 'Points, the members of
as its lines, and the containA
ing relation of E, as its incidence relation. Since, in the light of axiom (0), the
"
axioms (1), (2), (3) are merely the standard axioms of incidence for a 'Projective
plane, we see that every presentation is a projective plane.
In particular, then,
we may draw upon the various elementary consequences of the ax:1.oms of incidence.
example:
every member of
el
tinct members of
ep
(of
ei)
For
is contained in (contains) at least three dis-
e p ); if S, T are distinct members of el , then S /' T
of e; if A, S are members of e, e, respectively, and
p
p
A
(of
contains a unique member
if one of A, S is preassigned, the other may be chosen so that A ,;: S.
The following theorem shows that axioms (0) - (4) are not very restrictive:
Theore~
e
3.1.
If
n is a finite projective plane with
p1~~isely
v distinct lines,
and if F is a sk~w-field, then n has a prel?~t1:<2..~1) w:tth E a v-dimenolonal projectiva space ove! 1:.:.
Proof.
We shall actually show somewhat more,
system consisting
To
begin with, let n be merely a
of a collection of objects' called points and of a finite collec-
tion of v distinct pOint-sets, called lines, such that the following axioms are verified:
(i)
(ii)
Each ;point of
l!J,
Q
n is contained in at least one line of n.
are two distinct points of n, then the set of all lines of n con-
taining P is not identical with the set of all lines of n containing Q.
The effect of these aXioms is as follows:
..
If the lines of
n are numbered from
1 to v, then each point P of n corresponds uniquely to a non-empty subset of1, 2, ••• , v consisting of the numbers of the lines which contain P.
Clearly axioms
(i), (11) are satisfied not only by every finite projective plane but by a much
broader class of experimental designs.
Now we proceed as follows:
We represent E in form (2.1) where V is a (v + 1)-
dimensional vector-s:pace over F, and we select a basis for V:
We number the lines of
7f
from 1 to v and we assign to line i of
1C
the v-dimensional
subspace L of V defined as follows:
i
(3.3)
Ll = {f + el' e2 , ••• , ev }'
and so on.
MOre
L has a basis consisting of f + e i and the basis
i
Next, to each point P of 7f we assign a l-dimensional vector-
s~ecifically,
elements, e j' j ~ i.
subs:pace.
(3.4 )
uniquely determined by the requirement that i(l), i(2), ... ,i(k) are distinct positive integers which enumerate precisely those lines of
7f
which contain P.
11i is then
easy to verify that ptrCLj precisely when j is one of i(l), ••• , i(k).
This shows that
1C --
subject to axioms (i), (ii) -- always has a representation
subject to axiom (0), with incidence determined by the containing relation of E.
7f
happens to be a projective plane, then axioms (1), (2), (3) also hold.
If
And now
the proof of Theorem 3.1 is complete.
Theorem 3.1 suggests that there must exist a rich theory of linear representations of projective :planes, but it tells us little else.
A truly useful theory of
linear representations should exploit the theory of Desarguesian projective spaces
in order to throw new light upon the theory of :projective planes.
towards such a theory in the sections which follow.
We make a start
13
4. The normalization axioms. It will be convenient
to consider certain norma1i-
zation axioms which are not direct consequences of axioms (0) - (3) yet (to within
isomorphism) do not restrict the class of projective planes with linear representations.
These are as follows:
(N.1)
Each member of
ep
€g
is the intersection of' the members of
which con-
tain it.
(N.1')
Each member of
(N.2)
e -g is
the union of the members of
e;p
contained in it.
.
The empty pro jective subspace of E is the intersection of
er
The whole projective space E is the union of e .
_________________L
(N.2' )
Before stating a lemma concerning these axioms, we need a precise concept of
isomorphism.
By an isomorphism, ( S, ~), of a presentation
n(E,
ep' eI)
upon a presentation
we mean an ordered pair, (8, ~), such that 8 is a one-to-one mapping of
~ is a one-to-one mapping
eI
upon C
k'
,z.-p upon e~,
and the fo110uing condition is satisfied:
A c S precisely when A8 ~. sw , for all A in '
ep
and S in
By a smoothing isomorphism we mean an isomorphism
(e,
i) wIlIich satisfies two
(i)
e .
l
two additional conditions:
(ii)
It a member A of
members S of C:' l
members
~
'
ep'
p
is the intersection of a non-empty collection of
then AS is the intersection of the collection of' corresponding
•
(i1') It a member S of
A of'
6~
eI
is the union of a non-empty collection of members
then S~ is the union of the collection of corresponding members AS.
It n , 1(2' n are three presentations, and if K = (8, i) is a isomorphism of n
l
1
3
14
upon 1£2 and
K,.
«p.,
=
iR.f) is an isomorphism of 1£2 u:pon 1£3' then
isomorphism of 1£1 upon 1£3 and
K
= (a- l ,
are both smoothing 1somorphism then
= (aa t
KK I
1M' ) is an
,
l
iR- ) is an isomorphism of 1£2 u:pon 1£1.
is a smoothing isomorphism.
KK'
If K,KI
Our first lemma
gives a DNturaJ. exam;ple of an isomorphism.
Let (3.1) be a presentation, and let E' be a projective subspace of
Lemma 4.1.
E.
A necessary and sufficient condition tha.t the ma;EEing
X ->
(4.1)
X ....... E'
(where X ranges over the projective subspaces of E) should :i!'.:.2~.E:!! isomorphism of
the presentationJ3.l) u.pon a presentation
(4'.2)
is that, for all A in
e:p and
S in
el ,
if A r'I E' i9 contained in S, then A is con-
tained in S.
-
Proof.
lection
The ma:pping (4.1) induces a single-valued mapping
e; and
et
a single-valued mapping iR of
ep
a of
upon a collection
upon a col!- '
e. k '
where we
define
Aa =
(4.3)
for every A in
(3 p
A,,-.
and S in
SiR = S ~ E'
E' ,
e .
then, in Ilarticular, A ,.-, 1::' c
l
If
(a,w) is an isomorphism of (3.1) upon (4.2)
S implies A c S.
Conversely, let us assume that A,... 1::' c:. S implies A <:: S.
for some A in
contradiction.
e p'
then A,..... E'
tC
S, whence A
Therefore, the members of
C
e~
S, for every S in
If A ...... E' is emllty
e t.
(and hence the members of ~
non-empty and (tf course) finite-dimensional projective subspaces of E'.
--
are members of
ep
Aa = A.r'I E' c::: S
0"""\
and if S, T are members of
This is a
el
T and hence, by the condition, A
k)
are
I f A, B
such that Aa+ Ba c SiR "" ~, then
c::
S /'\ T.
Similarly, B
c:
S ,.., T.
15
As a consequence, if A ~ B, then S = Tj equiva'1ently"
if
S ~ T, then A = B.
This
is enough to show that eand ~ are one-to-one, that (4.2) is a presentation, and,
finally, that (a, ~) is an isomorphism of (3.1) upon (4.2).
And now the proof of
Lemma 4.1 is complete.
Although Lemma 4.1 is fairly obvious, it is qUite importantj on more than one
occasion the authors have drawn invalid conclusions by failing to check the necessary
(and sufficient) condition.
Our next lemma bears more directly than the
p~eeding
one u:pon the normaliza-
tion aXioms:
Le~ 4.2.
Every prese:ntE,+.ionJ3.l) has a!..~~::J.e s!n!J~~i:§. iSOl"1.orphis!1 u~
a presentation (402) wM.ch s~tisfies the four .E9..:s.malizati£E ~~~~N.ll-,jN.21 ).
.
Reme1"k~
~.:-~
,
As the proof of Lemma 4.2 will suggest, a pre3f.:ntat1.on (3.1) is likely
to have at least two essentially different smoothing isomorphi.sms upon presentations
subject to the four no::malization axioms.
Procf.
--->--
(a,
morphism
Let (3.1) be any presentation.
I) as follows:
I is the identity mapping of'
et
AS is the intersection of the members of
e~
=
eps,
e~ = e
,
I =
e,.
e ~ , e t)
e,
p
For each A in
We set E'
"
~l
=
e p'
E,
If a given member A of
el ,
then Aa= A.
ep
is the intersec-
Hence (ii) holds.
is the union of a non-empty collection of members A of
then, since A c Aa cS for each such A,
.
of corresponding members Aa.
-e
a smoothing iso-
We note tha.t, by defi:l1tion, A c: S if' and only
holds
S in ego Hence, the property (i) of an isomorphism /
is a prese:nt.s.tion.
If a given member S of
ing isomorphism.
et
which contain A.
tion of a non-empty collection of members S of
e..p ,
de~ining
"t "
if AS ~ S, for all A in
and n(E I
We begin by
S is also the union of the collection
Hence (iii) holds.
Since each member of
ep
I
Thus we see that (e, I) is a smooth-
has the form Ae, A in
e,
p
and since
16
e,; =
et
(5)~,
n(E',
'
it is clear from the definition of S· that the presentation
e k)
satisfies axiom (N.l).
It should also be clear that if n(l':,
satisfies anyone of the normalization axioms, so does n(E',
.
e pI,
e~).
~
Again let (3.1) be any presentation, and define (I,4i) as follows:
identity mapping of
ep •
which are contained in S.
c:.l'
For each S in
S~
ep ' et )
I is the
is the union of the members of
ep
This time, much as before, we may verify that (I,ili ) is a
smoothing isomorphism of (3.1) upon a presentation which satisfies (N.l').
Moreover,
if (3.1) satisfies anyone of the normalization axioms, so does the new presentation.
By iterating the two procedures in the two possible orders, we may define two
smoothing isomorphisms (Sl,i ), (S2'
l
which satisfy (N.l), (N.l').
A in
ep '
each S in
i1i
2
) of the presentation (3.1) upon presentations
The precise definitions are as follows:
A8 l is the intersection of the members of
el ,
S2
l
contain A, and, ff')r
e
where A ranges over the members A of
l
p
For each S in
, S~2 is the union of the members of
is the union of the AS
which are contained in S.
e p which are
e1 which
(a) For each
(b)
el
contained in S and, for each A in
e p'
AS
2
is the intersection of the
st 2 Where. S ranges over the members of .(;1 which contain A.
unless (3.1) itself satisfies one of
It should be clear that,
(N.l), (N.l'), the two smoothing isomorphisms
will have different effects.
At this point we may reduce our considerations, without loss of generality, to a
presentation (3.1) which satisfies both ot (N.l), (N.l'). Let Eo be the intersection
of
e g.
~
Then, in view of (N.l), E is also the intersection of
(N.11), if E is the union of
1
0
ep
I
tion of all members A of t3 p' considered as
to be the collection of all members S of
E'.
el'
Similarly, by
1/.
Now we define
e
then E is also the union of
l
.
E' to be the quotient projective space El/E (See Lemma 2.1) I
o
e.
P
ep
I
~
to be the co11ec;;'
projective subspaces of E', and
e t'
considered as projective subspaces of
It should be clear that the identity mapping of E induces a smoothing isomorph1
17
ism (6, ~) of (3.1) upon 1t(E I
,
ep " ell),
.
sUbject to all four normalization axioms.
and that the latter is a presentation
This completes the proof of Lemma 4.2.
Later we shall note the effect of the normalizing procedures just described upon
the presentation given in the proof of Theorem 3.1.
In what follows we shall consider additional axioms for presentations without
necessarily imposing the normalization axioms.
The latter will have their value in
simplifying proofs and constructions.
5.
The lattice aXioms.
with a presentation
Let us introduce the following notations in connection
('.1):
If A, B are distinct members of
e p'
then [A, Bl is the
unique member of
eI
containing A + B.
If S, T are distinct members of ~I' then [S, T] is the unique member of
e
ep
contained in S "" T.
The axioms (0) - (3) say almost nothing about the relation of [A, B] to A + B or
of [S, T] to S ...... T.
In order to intensify the relationship between the presentation
(3.1) and the "carrying" space E, one is tempted to impose one or both of the following lattice axioms:
(4)
(4 1 )
If
s,
T are distinct mem",)ers of
I f A, B are distinct members of
C?l
'
then [S, T] = S ........ T.
e p'
then [A, B] = A + B.
The aXioms (4), (4') are dual to one another.
..
of
We begin by studying the effect
(4).
Lemma 5.1.
I f (3.1) is a presentation subject to axiom (4), then there exists
a projective subspace E of E With the folloWing properties:
o
=
Eo for every two distinct members A, B of
e.
p
18
(ii)
_in
If A, S are members of
ep' el ,
re6pe.~vely, and if A is not contained
S, then,
A·
1"'"'0 S
= I: •
-0
Proof.
(iii)
We begin by proving an awd.liary proposition:
If A, B are distinct members of,
.ep
and if S is a member of
t
~ ~
= Art B.
contains A but not B, then S.""'I B
We note that (i), (11) certainly imply (iii)-and this explains why (Ui) is not
stated in Lemma 5.1.
To see that (iii) is true, note that [A, Fl:La a member of
which containa A and is distinct from S.
S '"'
=
However, [A, B],..,B
f:'t
Hence, by (4),
C4,. JB]
= A.
B, and, consequently,
which proves (iii).
Next consider three distinct members A, B, C of
ep •
If no member of
(3t
con-
tains all three of A, B, C, then, by (iii),
[A, B],.....C
=
and similarly with A, B, C permuted.
A '"' B
If some member, S, of
one member, D, of t!
p
eX
=
A """ C =
B r' C,
Hence, in this case,
=
A,..., C
B "'""' C.
contains all three of A, B, C then there exists at least
which is not contained in S.
Since S
= [A, B] and since
A, B, D are distinct, then, by the previous case with C replaced by D,
•
S."" D
=
A ~ D
2
B
r-
D :: A "
B.
This last equation must remain true when A, B, C are permuted.
for every three distinct members A, B, C of
Therefore (5.1) holds
e.p
Finally, let A, B, C, D be any four distinct members of
C' p. Then, by several
19
applications of (5.1),
= B.""'t C
Now it is clear that E
o
=
D
B-
=
eXists with property (i) c
to also has property (ii).
C ...... D.
And then we deduce from (iii) that
'!his completes the proof of Lemma 5.1.
'!he following lemma is clearly dual to Lemma 5.1 and hence requires no further
proof:
Lemma 5.2.
If (3.1) is a presentation subject
to aXiom (4'), then there exists
a projective subs:pace E of E with the following pro];lerttes:
l
(i)
S + T
= El
for every two distinct members S" T of
(ii) If A, S are members of
in S, then A + S
e p' e i '
eI
.
respectively, and if A is not contained
= El •
'!he folloWing theorem is probably known in a more general lattice-theoretical
formulation.
Nevertheless, it is a significant :part of the present investigation.
Theorem 5.3.
If the presentation
Jt
= Jt(E,
e p ' e l)
satisfies both of the lattice-aXioms (4), (4'), then the projective plane Jt is
Desarguesian.
Sketch of proof.
In view of Lemma 4.1, we may assume without loss of generality
that the normalization aXioms are also satisfied.
Eo empty, and Lemma 5.2 holds with E
l
2t-l
It follows rapidly that, for some positive
has (projective) dimension t-l, each member of ell
.
~
and the whole space E has dimension 3t-l. At this point we use
integer t, each member of
has dimension
= E.
In this case Lemma 5.1 holds with
ep
the vector-space representation (2.1) and prove the theorem of Desargues for Jt by
-e
vector-s:pace methods.
We may safely omit the details, inasmuch as a similar but
20
somewhat more complicated proof must be given later.
(See Theorem 6.1.)
We shall omit examples of Theorem 5.3 until we have introduced certain "completeness" axioms.
Our next theorems will show that aXiom
little restriction upon presentations.
tion.
(4)
or
(4'),
taken alone, places very
It will be convenient to begin with a
de~ini-
If n is a positi ve integer and if F is a skew-fi.eld, let
be the least positive integer d such that a d-dimensional projective space E over F
2
contains at least one set of n + n + 1 distinct points having the property that each
subset of 2n + 1 distinct points, chosen from the set of n2 + n + 1, is linearly
This means that if 1 ~ k ~ 2n + 1, each k of the points have as their
independent.
e
union a projective subspace of E of dimension k - 1.
2
(5.3)
2n ~ d(n, F) ~ n + n
for all choices of n and F.
d(n, F) for the case F
Obviously
Moreover, if, for q a prime-power, d(n,q) denotes
= GF(q),
it is easy to see that
d(n, q) = 2n
for all sufficiently large q.
For example, d(2,q)
d(2, q) = 4 for every other prime-power q.
Theorem
::r
5 for q = 2, 3, 4, 5 but
Now we are ready for a theorem:
5.4. Let n be a projective plane of finite order n, let F be a skew-
field, and let d(n, F) be th..:.....~r.l'teger defined above. .',rhen
which satisfies axiom
1C
has a preeentayion (3.1)
(4) and the four normalization axioms (N.l)-(N.2') and has the
following additional properties:
(i)
(ii)
(iii)
E is a d(n, F)-dimensional projective space over Fe
Each member of
Each member of
e
is a point of E.
p
eI
-
is an n-dimensional projective subspace of E.
21
Remarks.
1.
In an earlier form of the present theorem, the dimension, d(n, F),
of E was replaced by n
2.
2
+ n.
The improvement was suggested by Da.le Mesner.
It seems likely that d(n, F) = 2n whenever F has sufficiently many elements.
Simple counting erguments show that this is true when F
Proof.
= GF(q).
(See below$)
It will be convenient to define
v
= n2
+ n + 1,
k
=n
+ 1.
of
We choose E to satisfy (i) j'lneorem. 5.4.
Then, by definition, there exists at least
one set
(5.6)
P , P , ••• , P
v
l
2
of v distinct points of E having the property that, for 1 <
= k < 2n + 1, each k of
::l
these points generate a projective subspace of E dimension k - lover F.
to be the collection whose v members are the points
e
ep
(5.6).
Next we enumerate the v distinct points of :re as 1, 2, ••• , v.
of:re we make corresllond the i th member, Pi' of
We take
ell.
To the i th point
To each line L of :re there cor-
resllonds a unique set, J(L), of k distinct llositive integers determined by the
requirement that i is in J{L) if and only if the i th point of :re lies on L.
To L we
make corresllond the n-dimensional projective SUbspace, L'"of E generated by 'the
k
=n
+ 1 distinct lloints Pi' i ~ J(L).
And we define
tt to be the collection whose
members are the L', where L ranges over the v lines of E.
It should be clear at once
that
so defined, is a presentation of:re.
Now consider two distinct lines L, M of :re, and
note that
dim(L' "" M')
= dim L' + dim M' - dim (L' + M' )
= 2n
-e
-
dim (L 1 + M').
22
Now L, M intersect in a unique point of
1C,
say the i th point.
Hence J(t) """""J(M) con-
Sists of precisely 2n + 1 distinct integers, and t' + M' is generated by 2n + 1 distinct members of
t3..;
p
that is, of (5.6).
=
dim (L'"..... M')
Hence
2n -
2n = 0 ..
Therefore L',..... M' is a point, namely the point Pi.
This shows that the presentation
(5.7) satisfies axiom (4).
the union of
ep
is Ell and if E has dimension less than den, F), we have
l
a contradiction to the definition of d(DI F) ,Hence E ,.; E. Thus axiom (N.2') holds.
l
By the definition of e I" axiom (N.1') holds. Since axiom (4 ) holds, and since the
If
members of
C'p are points, then axioms (N.l), (N o 2) hold.
This completes the proof
of Theorem 5.4.
In ad-dimensional Desarguesian space E, a valid theorem regarding projectives
subspaces of E remains valid when.. in the statement of the theorem, an s-dimensional
subspace is replaced oy"a
int~rsectionD are
Theorem 505.
(d~a~l)~dimensi~nal subspace
interchanged.
Let
1C
for eash
5} andu~ons ~~d
Applying this duality to Theorem 5.4) we get:
be a projective plane of finite order n, let F be a skew-
field, and let den, F) be 'the same integer as in Theorem 5.4.
Then
1C
has a presenta-
tion (3.1) which satisfies axiom (4') and 'the four normalization axdoms and has 'the
folloWing ed<"'1t1onal properties:
(i)
(ii)
E is a den, F) - dimensional projec'tive space over F.
Each member of
et
1s eo bNPerplane of E,
"..
-.
is a projective Bubspace of E of dimension.
_
- ~ ..:(..:Ui)
.
den,
t) -
Each member of
"-"'
ep
n - 1 over F.
Just as in the case of Theorem 5.4, the original form of Theorem 5.5 had d(n,F)
23
replaced by '02 + n.
In this weaker form, Theorem 5-5 was obtained from the construc-
tion given in the proof of Theorem 3.1 by applying the normalization techniques (see
the proof of Lemma 4.2). Then Theorem 5.4 (in its weaker form)was obtained by duality_
It is quite possible that every projective plane
1C
sentation (3.1) with £ of dimension less than d(n, F).
of finite order
'0
has a pre-
One possible method of proof
would be to start from the presentation (3.1) given in Theorem 5.5 and then attempt
to apply Lemma. 4.1.
In view of Theorem 5.3, we can impose at most one of axiloms (4), (4 1 ) if we hope
to construct presentations of non-Desarguesian projective planes.
Since (4), (4 1 )
are dual axioms, it matters very little which of (4), (4 1 ) we decide to investigate
further.
We choose to retain axiom (4).
In view of Theorem 5.4, it seems desirable
to add further axioms as well.
Before leaving this section we shall ske tch a rough proof of (5.4).
a
=
2
'0
+
'0,
b
=
Set
2'0
and let q be any prime-power such that
q
Let £
= PG(2n,
>
=
t:~)
'lb
•
q) be the projective space of dimension Bn
<
that, for some t in the range b + 1 = t
=b
over GF(q) and suppose
<
=a, we .have found a set T of t distinct
points of £ each b + 1 of which are independent.
Then each set of b points in T
generates a hyperplane of £, end the number of hyperplanes so generated is at most
Therefore, there is at least
p01nt P of £ which is in Mne of these hyperplanes.
But "chen the set T w P consists
24
of t + 1 district points of t, each b + 1 of which are independent.)
mathematical induction, we get (5.4). for q can be
:Lm:Pl'~ed
Therefore, by
It should be obvious that the lower bound
greatly by a more refined a.na.lysis.
6. Rigidity. The follOWing definitions will be useful in connection with a
presentation (3.1):
A member, A, of
tinct members S, T of
[S, T]::
ep
et
is rigid (With respect to
et ) if,
such that at least one of S, T contains A, it.. is true that
S ..-"I T.
e1 is rigid (With respect to e p)
B of ep such that at least one of A,
= A + B.
A member, S, of
tinct members A,
true that [A,B]
if, for every pair of disB is contained in S, it is
Note that axiolll' (4) can now be restated as follows:
rigid.
for every pair of dis-
Similarly" axiom (4') can be restated as:
Every member of C
p
Every member of
Since" in the present paper, we intend to retain axiom
(4),
e1 is
is.
rigid.
the concept of rigidity
will mainly be of interest in connection with members of ~1.
For the next theorem we need the geometric definition of a translation plane.
A
projective plane 1( is a translation plane with respect to a line L of 1( if Desargues'
theorem holds whenever the center of perspectivity is a point of L and the axis of
perspectivity is the line L itself.
A very deep theorem states that if a projective
plane 1( is a translation plane with respect to two different lines then 1( is a translation plane with
re~pect
to every OLe of its lines.
In this case 1( is a Moufang
plane (a plane coordinattzed by an alternative division ring).
Marshall Hall [4].
We may add the following:
For more details, see
Le1l 1( be a projective plane" L be one
of its lines" and 1(1 be the affine plane obtained from 1( by deleting L and its points.
Then 1( is a translation plane with respect to L precisely when 1(1 permits a certain
e
25
principle of parallel propogation of triangles, frequently called the Little
Desargues' Theorem.
In Bruck [5] this principle is called the Vector Axiom.
The following theorem together with its corollary, shows that, for a presentation
(.3.1) subject to s.x1om (4), the eXistence of one rigid member of
C!{ implies that the
corresponding projective plane is a translation plane; and the eXistence of two distinct rigid members of
C't
implies that the plane is Desarguesian.
The fact that
Moutang planes do not enter eX1?licitly shows clearly that rigidity is a property of
revresentations and not of projective planes.
Theorem 6.1.
If the Eresentation
(6.1)
1t
I:
1t(E,
t)
ep '
C9
satisfies the lattice-aXiom (4) and if some member, R, of
projective plane
1t
is a translation plane with reepect to the line R of
addition, some two members A, B of
:.t
members. gf
Remark.
lations.
1t.
If, in
ep , exactly one of which is contained in R,
the same dimension. then aXiom (4') holds and
Corollary.
c?, is rigid, then the
1t
have
is DesaraHssian.
If the presentation (6.1) satisfies aXiom (4) and if two distinct
are rigid, then
1t
is Desarguesisn.
'lb prove the Corollary we need only verify axiom (4') by simple calcu-
There is no need, for example, to appeal to the deep geometric theorem
about Moufang planes mentioned above.
Let us call a member, A, of e ideal or affine according as A is con.
11
tained in R or not contained in R. And let
* be the collection of affine members
Proof'•
ep
of
ep •
Again, let us call R the ideal member of
affine members, and
system
--
c: 1
j
all other members of
et* denotes the collection of affine members of ef.'
c:. f.
are
Then the
26
is the affine plane
* obtained
1(
In view of Lemma
from
1(
by 1e1eting the line R and its points.
4.1 we may assume Without loss of generality that the normali-
zation axioms are ~atisfied. Then, by Lemma 5.1, eve~y two distinct members of
are disjoint (have no common points); indeed, each member A of C?
.
every member, 8, of
C:t
p
Which, does not contain A.
c:p
is disjoint from
We use these facts in computing
(projective) dimensions as follows:
The ideal, member, R, of
A, B, C of
e.
p
<='t
contains at least three distinct (ideal) members
By rigidity, R = A + B.
Since A,.., B is empty, and hence has dimen-
sion -1,
dim R
=
dim A + dim B - (-1)
+ dim B
= dim A
The same is true with A, B.J C permuted.
+ 1.
Hence, there must eXist a positive integer i
such that
dim A =
dim R
=
- 1,
i
2i - 1.
Next let A, B be two (affine) members of
ep ,
[A, B]
set
[A, BJ ~ R
C =
Then C 1s an ideal member of
ep * and
and
=
A + C
=
B + C
since R is r1gid..The't'e£ore
dim [A, B] =
dim A + dim C + 1,
and the same 1s true with A replaced by B.
a
Hence there must eXist a positi ve integer
such that
(6.4)
dim A = a
(6.5)
dim 8
= i
-
1,
+ a
A E
-
1,
8 E
27
,r-*
U
C
Again, if A, B are two distinct members of
p
,
our formulas yield
dim [A, B]- dim (A + B) = i - a .
Therefore
(6.6)
•
>
= a,
i
with equality precisely when axiom (4') holds.
This explains the final sentence of
Theorem 6.1 (and also explains the Corollary.)
.0* 1 then
As preViously shown, if A, B are distinct members of ~p
[A, B]
where C c R.
=
A + C = B + C
Consequently,
[A, B] + R
=
A + R = B + R.
From this and the normalization axiom (N.2') we may conclude that
(6.7)
e
By
A+R
(6.7)
A
E,
and the fact that A . . . . R is empty, together 'With
(6.8)
dim E
In addition,
(I)
=
(6.7)
=
€
e*
p •
(6.3), (6.4),
a + 2i - 1.
implies the following proposition:
If the projective subspace Wof E contains at least one member of ~p * , then
(6.9)
W + R
=E,
dim W = dim (W ~ R)
(6.10)
+ a.
Note that to get (6.10) we use the formula
dim W + dim R = dim (W + R)
together with
(6.9), (6.8)
and
+ dim (W /"0 R)
(6.3).
The form of Desargues' Theorem which we need for the proof of Theorem 6.1 may be
stated as the following proposition:
(II)
Let A, B, C, A', B', C' be six distinct members of t>p* and let 0, C", A"
28
be three distinct ideal members of C'p such that (i) [C, A] does not contain
° and
(11) the following equations hold:
(6.11)
[A, At]
~
R
[B, Bt ] " R ::&
::I
[C, C']~R
::I
(6.12)
[At B] n R
::I
[AI, B'],., R
::&
C"I
(6.13)
[B, C] ~ R
::I
[B , CI],.... R
'
:=I
A" •
Then there eXists an ideal member, Btl, of
(6.14)
0,
e P such that
B".
[C,A]/,,\R::I [C',A']"R::&
We begin by concentrating on A, B, At, Bt ,
Proof of (II).
° and c".
(We shall
derive, without explicit formulation as a proposition, a crucial result involving
e p .)
these six members of
Let us define
(6.15 )
(6.16)
Z
::I
°1
::I
C
l
::I
+ B' ,
(A+A').-R,
°3
(6.17)
A + B + AI
°2
:;
(A + A')
/"'I
C
2
=
(A + B) _
(B + B') ,.... R,
=
(At +B t ) -. R,
(B + B'),
(A + B) ..... ,R,
C
3
=
(AI + B' ).
Clearly
03 c [A, A']/'\[B, B']
OcR,
:=I
by (6.11), and hence
03
::I
°3
/"'1
R =
(A + A' ),,-. R
r'\
(B + B') ;""\ R::I 01 """ 02.
Thus and similarly,
(6.18)
03
=
01 r, 02'
By proposition (I), together With (6.4),
dim 01
Thus and similarly,
(6.19)
::I
dim (A + AI) - a
::I
a - 1.
29
From (6.15), (6.16), (6.18),
dim Z
=
dim (A + A') + dim (B + B') - dim
(°1 - °2 ) •
Applying proposition (I) three times to this, we get
dim (Z .,.... R) + a = dim 01 + dim 02 + a
= 2a
•
- dim (01 '" 02)
+ dim (01 + 02) •
Thus and similarly,
(6.20)
If
~ = 01 + 02 + C1 + C2 '
(6.21)
then, obviously, by (6.16), (6.17), (6.15),
(6.22)
Again,
Thus and similarly,
(6.23)
01 + 02
1 + C2 .::: C".
Since 0". CIt is empty, we see from (6.23), (6.21) that
c.
0,
C
(6.24)
Now let us set
(6.25)
= a + k - 1
dim (Z /"'\ R)
so that, by (6.20),
(6.26)
dim (01 + 02)
=
dim (C 1 + C2 )
We note from (6.26), (6.19) that
k - 1
~
dim 01
=a
or
(6.27)
k
>
= a.
- 1
=k
- 1.
30
On the other hand, by (6.25), (6.22), (6.24), (6.26),
k +a - 1
whence a >
= k.
>
=
dim Zl = 2k - 1,
By this and (6.27),
(6.28)
k = a.
By (6.28) in (6.26), and by (6.19), 01 + 02 has the same (finite) dimension as
<\
..
-,
or O2• Therefore 01 = 02.
(6.29)
01
By this and (6.18), we may write
02
Q
=
03
*
= 0.
Similarly,
*
C1 = C2 = C = C.
3
Moreover, by (6.28) in (6.25), and by (6.28), (6.25), (6.24),
(6.30)
Z ....... Rand Zl have the same dimension 2a - 1.
Thus, by (6.22), z,.., R = Z••
Now we use (6.29), (6.30) in (6.21) and get
(6.31)
=
Z r. R
0* + C*.
In addition, by (6.29), (6.30), equations (6.16), (6.17) simplify to
(6.32)
(A + A' )
(6.33)
(A + B),..... R
r"'\
R
= (B
= (A'
+ B') ,..... R
+ B')" R
= (A + A') r' (B + B' ) =
= (A + B) I'" (A' + B') =
0*,
C*
where
(6.34)
(6.35)
0*
c:.
=
dim 0*
°,
dim C*
C*
=
c
C,
a - 1.
Equations (6.31) - (6.35) constitute the "crucial result" to which we referred
at the beginning of the proof of :proposition (II).
0,
~'
(6.36)
then
If now we r'eplace A, B, A', B',
by B, C, B', C' , 0, A" respectively, we may conclude on like grounds that if
x
=
B + C + B ' + C'
31
(6.37)
=
X,.,R
(6.38)
(B + B')r, R
= (C
(6.39)
(B + C),.. R
= (B'
+ A*,
0*
= (B
C') r. R = (B
+ C')",R
+
+ B');'"\ (C + C')
+ c) ,.... (B' + C')
=
=
0*,
A*,
where
(6.40)
0*
(6.41)
dim 0*
c
0,
A* c: A,
=
dim A*
=
a - 1.
Note that, because (B + B')" R occurs both in (6.32) and (6.38), 0* has the same
meaning in formulas (6.36) - (6.41) as in formulas (6.31) - (6.35).
Next we define
(6.42)
(C + A)
B* =
r\
(C' + A').
We shall use vector methods to prove that
(6.43)
B*
c
c*
+
A*.
Before embarking on the proof of (6.43) let us note that (6.43), together with
(6.42), (6.40), (6.34), implies that
B*
c;
[C, A] .,., [C', A'] ".... R.
Hence, if we know that (6.43) holds and that B* is nOll··empty, we will be able to conclude that (6.14) holds for some ideal member, B" I of
tains B*.)
eI'
(where, of course, B" con-
Therefore, we may end our proof of (II) (and of Theorem 6.1) with the
proof of (6.43).
By (6.15), (6.36),
(6.44)
Z + X = A + B + C + A' + B' + C' ,
Consequently, all of the projective subspaces A, B, C, A', B', C', 0*, A*, B*, C* are
contained in Z + X.
Moreover, with the possible exception of B'*') all of these sub-
spaces have projective dimension a - 1.
32
Next we need:
(6.45) Each of A + B, B + e, e + A is disjoint from 0*.
We see this as follows:
Since, by hypothesis, [e, A] is disjoint from 0, and since
e + A, 0* are contained in [e, A], 0, respectively, then e + A is disjoint from 0*.
On the
ideal
A+B 'is disjoint fr{)I!l 0* em 'the grounds that 0, en.
o-ther haind,
~;rs
from' 0*.
t·
p
of
And B+e
&1d that ,fA, 13-] ctm-ta1ns
Ct ! and
~e~e
{W6'
distin<it
must be d1sj'oint
is disjoint from 0* on simila.r grounds.
At this point, we represent the projective space Z + X as a vector-space over the
skew-field F.
We begin by selecting a fixed but arb!trary basis for 0* over F:
(6.46)
We note that 0*, A, Al are a-dimensional vector subspaces over F, that each two intersect in the zero vector-space (O}, and that
0* + A
= 0*
+ AI
=A + A
I •
Hence (coffi:?are the similar remark in Bruck and Bose [lJ ) there exists a unique basis
of A over F such that
(6.47)
... , a ),
a
Similarly
(6.48)
... , ba L
= (b l ,
B'
el
...
O*+A+B
must have dimension 3a.
= (e l
= (e l
= 0* + AI + B'
Since, further,
A*
=
(B + e) ....... (B '
+ Ci
),
+b ,
l
... , e a
+ b },
a
, e a + ca J.
, c a },
e = (c '
+ cl '
l
Since, by (6.45), B + e is disjoint from 0*, the vector-space
B
...
it is clear from
(6.48), (6.46) that
...,
b
a - c}
a
.
Similarly
(6.50)
B*
=
c* =
{c l - al, .. ·,ca - bal
fa l - b l , ... , aa - b a } •
Moreover, each of A*, B*, C* has dimension
from
(6.49), (6.50)
(6.51)
,
"a
as a vector-space.
It is clear
that
A* + B*
= B*
+ C* = C* + A* •
This is more than enough to complete the proof of
(6.43),
proposition (II) and
Theorem 6.1.
At an earlier point we merely sketched the proof of Theorem 5.1.
needed to prove the unrestricted form of Desargues' Theorem.
There we
However, the essence
of the omitted proof (after some preliminary calculations concerning dimensions)
is a set of equations analogous to
(6.46) - (6.51).
It will be convenient to sum up some of the details of the proof of Theorem
6.1 in the following lemma:
Lemma
6.2. If
a~resentation
(6.52)
satisfies axiom (4)
and the normalization axioms, and if
R, then there exists an ordered pair
.E.ropertie s:
(i, a) of integers
~
has a rigid member
i, a
with the following
> a >0
(a)
i
(b)
E
.
has dimension
a + 2i - 1.
has dinension
2i - 1.
(c)
R
(d)
Each member of
A member,
distinct
from R, bas
a + i - 1.
dimension
(e)
'
~.e
of ~p
A,
according as
has dimension
i-lor
a-l
A is contained in R or not con-
tained in R.
The question of the possible relationships between the integers
an interesting one.
According to Theorem 6.2, the case
Desarguesian planes.
i.
Later we shall give
a
=i
i
and
a
is
occurs only for
"natural" examples in which
a
divides
The following lemma shows that a given projective plane may be associated with
a range of values for
LeI!1E!:~>.6.:2.
If
a:
a projective plane
1t
fies the hypotheses of Lemma 6.2 with a
has a presentation (6.52) which satis-
> 1, then
which satisfies the hypotheses of Lemma 6.2
1t
has another presentation
with the pair (i, a) replaced by the
pair (i, a-l).
!3:2Qf. Let E' be any fixed hyperplane of E
the set of all projective subspaces A _ E', A
of all projective subspaces
(6.53)
1t (
S '"'" E' ,
e'p'
E ,
is a presentation with the
S
€
e.e'
which contains
€
ep '
and let
R.
at
1J'3t
a'p
be the set
We shall prove that
-€')
J,
desired properties.
For the proof we shall use the
notation and reSUlts of the proof of Theorem 6.1.
.-
If A is in
of
.e*,
P
then A + R
=E
E containing at least one member of
For any W not contained in
E',
W+
.
Hence, if W is a projective subspace
.e*p I
then
E'
= E
W is not contained in
and
be
35
dim(W _E')
=
dim W +
=
dimW-l.
=
a - 2 ,
dim
E'
-
dim E
In particular,
(6.54)
dim(A .... E' )
(6.55)
dim( 8 _ 1: ) = a _ 1 + i - 1 ,
'
Now suppose that A, 8
is non-empty, then
Theorem 6.1,
A,..... 8
is non-empty.
A is contained in
is indeed a presentation.
[A .- E',
If
(A"...E') '" (8 _ E')
But then, as shown in the proof of
Indeed
=
B _ 1:']
for every two distinct members
[A, B] ,.." E'
A, B of
[S_ E', T -. E' J
p
respectively.
From this it follows rapidly that (6.53)
8.
for every two distinct elements
--e*
-e .e
are in.Qp'
-ep ,
and
8 ...... T _
=
[8, T] ....... E'
=
(8 ~ E') ........ (T .-. E')
S, T
of ·e
.e
.
=
1:'
In addition, if A,
C
are in
-ep - -e*,
re spectively, then
p
=
(A + C)
r'I
E'
=
(A .-. E' )
+
C,
the last step following, for example, by comp;:;':d.son of dimensions.
These remarks should suffice for the proof of Lemma 6.3.
marking that, although the mapping
X
->
X /""\ E'
It seems worth re-
(where
X ranges over the projective subspaces of
of the presentation
(6.52) upon the presentation (6.53), this mapping does not
induce a smoothing isomorphism. of
(6.52) upon (6.5'). What is much more likely is
that there exists a smoothing isomorphism. of
7.
E) induces an isomorphism
Completion axioms.
(6.53) upon (6.52).
The following coumletion Sioms are of interest in
connection with a presentation (3.1):
(5)
Each point of
(6)
If A, 5
1:
is contained in exactly one member of
1 '
are members of ~p'
•
-e •
p
respectively and if A ...... S
is non-
empty, then A c S.
We can, of course, write down dual axioms (5'), (6'), but we shall omit these.
Lemma.
==-:--
7.1. If a presentation (3.1) satisfies axiom (5) and one of (4), (6),
then the presentation satisfies all three of (4), (5), (6) and all of the normali_
zation axioms.
Proof.
Let
====
be a presentation satisfying axiom (5).
(3.1)
then, if A, B are distinct members of .. ,
p
Suppose further that
B are members of -ep
A __ S
=
and S is a member of
A r. B, which is empty.
and let
P
(unique) member,
(6),
of
A c; S ..... T.
-e.
p
Therefore
of
..e.
p
However,
holds, let
Then A _
S_ T
(8, T] = S _ T.
--e J
which contains
Hence axiom (6)
be a point of 8" T.
A,
A _ B is empty.
(3.1) satisfies axiom (4). Then, by Lemma. 5.1, if A,
If, on the other hand, axiom (6)
-e J
In particular,
holds.
5, T be distinct members of
By axiom (5),
S ""' T
P
is contained in a
is non-empty and hence, by axiom
cofttains one and only one member,
-_
(4)
We note that
implies axiom (N.l), that axioms (~), (5) imply axiom (N.2), and that
axioms (5), (6) imply axioms (N.l') and (B.2').
Lemma 7.1.
(8, T],
That is, axiom (4) holds.
At thi s point we may assume all three of axioms ( 4) , (5), (6) .
axiom
B but not A,
This completes the proof of
37
~~~=1=g.
E'
and let
Let
(3.1) be a presentation satisfying axioms (4), (5), (6),
be a projective subspace of E having the property that
(:.p.
is non-empty for every member of A of
- > X'
X
(7.1)
=
A
E'
Then the mapping
X,...., E'
(where X ranges over the projective subspaces of E) induces an isomorphism of
the presentation (3.1) upon a presentation
E' ,
1t (
e; ,
e.t' )
which also satisfies exioms (4), (5), (6).
Proof.
of form A
of form S
Here, of course,
'"'
E' ,
Ae
,... E' ,
If A ...... E' c
A !"""' E'
S
€
~;
~, and
p
is the collection of all projective subspaces
~,
.t
is the collection of all projective subspaces
e.t.
S, where A, S are members of
is non-empty and, therefore, by axiom (6),
t:>p'
A
~.t respectively, then
C
S.
Consequently, by
Lemma 4.1, (7.2) is a presentation and the mapping (7.1) induces an isomorphism of
the presentation (3.1) upon the presentation (7.2).
is non-empty, then A
C
Similarly, if
and hence A /"\ E' c:: S ~ E'.
S
(A"E' )..--(S"", E ,)
Therefore (7.2) satis-
fies axiom (6).
If P is a point of
axiom
E'
then, as a point of
(5), in exactly one member,
least one member of
e p'
A
namely in
of
A __
e.
p
E' •
E ,
P is contained, by
Therefore
P is contained in at
If B is a member of
tinct from A then P is not contained in B or B ......, E'.
fies axiom (5).
In view of Lemma 7.1, the proof of Lemma 7.2
~
p
dis-
Therefore (7.2) satisis now complete.
The following theorem is intuitively obvious from Lemma 7.2, but nevertheless
--
requires proof:
38
~~g~~~_1=~.
satisfies axioms
If a pro,iective plane:n:
(4), (5), (6)
then
:It
has a presentation (3.1) which
has a presentation (3.l) which satisfies
not only axioms (4), (5), (6) but also the following requirement: __ To each hy;perplane
~p
such that
This means, in particular, that various members of
are
of I:
I:'
there corre sponds at least one member, A, of
is empty.
points.
~~gg!.
(By Zorn's Lemma and Lemma 7.2)
Let ~ be a non-empty collection of
projective subspaces of
I:, linearly ordered by inclusion, having the property that
if L is in ~ , then
L '"' A is non-empty for every member A of
Lo =
all spaces
Consider a member,
L __ A, L
~.
£. .
ep
A, of
The collection
and the collection
;;e .......
~.
p
;e.",",
Let
A
of
A consists of non-empty pro-
jective sUbspaces of the finite-4imensional projective space A and is linearly
ordered.
Hence
£- "'"'" A is finite and has
though Lo;is not assumed to ~ in ';,( .)
member A of
space
C p ..
L ""'" A as its least member (even
o
Therefore L ("'\. A is non-empty for every
o
Consequently, by Zorn's Lemma, there exists a projective sub-
of Z with the following tvo properties:
o
(i) A" I: is non-empty for every member A of ~.
o
P
(ii) If
I:' is a projective subspace of I: which is properly contained in
I:
I: ' then
o
A '"' I:'
is empty for at least one member A of
Applying Lemma 7. 2 (with
Theorem 7.3.
- Note that if
I:'
:n:
replaced by
~p'
I: ) we get the conclusion of
o
is assumed to be finite or (more generally) if
is assumed to be finite-dimensional, Theorem 7.3 follows directly from Lemma 7.2
without the aid of Zorn's Lemma.
-e
I:
39
8.
Some presentations of Desarguesian planes.
We begin by recalling some
facts about skew-fields and Desarguesian projective planes.
The center,
ck
= kc
a field.
C, of a skew-field K is the set of all c
for every k in K.
in K such that
And C is not only a sub-skew-field of K but also
By a theorem of Kaplansky
[2], a necessary and sufficient condition that
a skew-field K have finite dimension over its center C is that K satisfy a
non-trivial polynomial identical relation.
Next let
K.
be a Desarguesian projective plane coordinatized
~
First let us suppose that
by
a skew-field
~
satisfies a (universal) configuration theorem
it
which is non-trivial in the sense thatl is neither a consequence of Desargues' Theorem nor equivalent to the assertion that K has characteristic p
p.
Then (see Marshall Hall [3])
relation.
then
~
for some prime
K satisfies a non-trivial polynomial identical
Conversely, if K satisfies a non-trivial polynomial identical relation,
satisfies a non-trivial (universal) configuration theorem.
marks, coupled with the
above-mentioned theorem of Kaplansky, a necessary and
sufficient condition that
theorem is that
By these re-
~
satisfy a non-trivial (universal) configuration
K have finite dimension over its center
C.
Now we are ready for
our first example.
~~~;=~~~~
Let
K be a skeW-field which has finite dimension,
somee SUb-field F of its center.
coordinatized by K.
Let
In the sense of
~
=E
~
t, over
be a (Desarguesian) projective plane
(2.l) we may represent
~
in the form
(W/K)
where
W is a 3-dimensional vector space over K as a ring of left-operators.
Since
K is t-dimensional over F and since F is contained in the center of K,
we may represent W as a 3t-dimensional vector-space V over F with the
-e
following properties:
40
(i)
V admits
K as a skew-field of left operators .
(ii) For 0 ~ s ~ 3, the s-dimensional vector subspaces of W over K are
precisely the
st-dimensional vector subspaces of V over F which are
K-spaces
in the sense that they are mapped into themselves under left-multiplication by
elements of K.
Now we define
E (v/F)
=
take
~
to
p
F which are
,
the collection of all t-dimensional vector subspaces of V over
1:B
K-spaces,
e.e
and take
to be the collection of all 2t-dimensional
vector sUbspaces of V over F which are
1t (
K-spaces.
Then, as is easily verified,
e p' e.e )
E ,
for Desarguesian planes may be studied in new forms in higher dimensional space over
a field.
Using this example, and choosing
such that
A,- E'
presentations of
(5) and (6)
1t
-
via the method of Lemma 7.2
(4').
of
~R
-
e
we
may derive new"
which satisfy axioms (4),
We shall treat two oases.
We start from Example 8.1, so that the projective space
has dimension 3t-l, the members of
members of
to be a projective subspace of E
is non-empty for every member A of
but not always
~~~e 8.~=
E'
~
p
E
have (projective) dimension t-l and the
have (projective) dimension
2t-1.
We single out some member,
1:::.B (and we observe, in passing, that the members of
e-p
R,
contained in R con-
stitute a spread of R in the sense of Bruck and Bose (1).) And, finally, we choose
-e
a 2t-dimensional projective subspace,
E' ,
of E which contains R.
Then
41
R is a hyperplane of
A ,...... R is empty,
E'.
A+
l': p not contained in R, then
If A is a member of
E'
=
=E
A + R
= dim A +
dim(A """" E')
,and
dim
dim E
E'
= (t-l) + (2t) - (3t-l) = o.
Hence A ~ E'
R.
~
is a point for each member A of
p
which is not contained in
Now it is easy to verify that
:It (
l:'~,
E "
~t )
,
constructed in the manner of Lemma 7.2, is a presentation of the Desarguesian plane
which has precisely the form given in Bruck and Bose [1].
:It
satisfies axioms
(4), (5), (6). In addition, R is a rigid member of
Moreover, in the notation of Lemma 6.2, we have
that, had we taken the projective subspace
2t + k instead of 2t, where
tation with a
The presentation
=k
+ 1,
i
0 <k <
=
E'
i
= t,
a
= 1.
It may be added
containing R to have dimensi. on
- 1, we would have arrived at a presen-
~rt
='
= t.
Before com:ing, to the next example we wish to have a lemma.
if t
is a positive integer and
q is a prime-power, then E
We recall that
= PG(2t-l, q) is
the projective space of dimension 2t-l over the finite field GF(q).
a spread .,
-8,
of
E is a collection of
Moreover,
(t-l)-dimensional projective subspaces
of E having the property that each point of E is in one and only one member of
~
The number of members of ..& is
Lemma 8.3.
Let
...8
projective subspace of
1 < i < k, let
= =
0_
A ....... E'
~
be a SEread of
qt + 1.
E
=
PG(2t-l. q)
E of dimension 2t-k where
and let
2 < k < t.
= =
E
I
be a
For
be the number of distinct members A of"g such that
has dimension t-l.
Then we may draw the following conclusions:
42
(a)
If 1 ~ i < (k+l)/2
(b)
If l<i<j<k and i + j
(c)
The equations
k
t
E
xi = q + 1
i=l
k
t
k-i
k-l
E
q
xi = q + q
i=l
=
(8.1)
(8.2)
, then
xi = 0 or l.
=
~
k, then XiXj = O.
hold.
If k
~g&~~~~=~.
2!.
(A)
~
(B)
xl
~~gg~.
= 2, t
~
~
If k = 3, t
2, then xl = 1, x2
3, then either
= 1, X2 = 0, x3 = qt ,
t
= 0, x2 = q + 1, X; = q
- q •
We begin by assuming that, for integers
i, j
1 ~ i ~ j ~ k there exist distinct members A, B of
B n
E'
have dimensions
= qt •
t-i, t-j
respectively.
A
such that
such that A,,", E',
Then, since A, B have no
common points,
(A .- E')
+
E'
is a projective subspace of
(B ...... E ')
of dimension
(t-i) + (t-j) + 1
= 2t + 1 - (i + j) .
therefore we must conclude that
i+j>k+l.
When we interpret this result for the cases
i
= j,
i < j, we get
respectively.
To prove
(c), consider a member,
w
=
dim (A +
A, of ~ and set
E').
(a), (b),
43
Then
dim (A ,-... E' )
dim A +
=
dim
-
= (t-l) + (2t - k)
and
dim (A
~
E')
= t-i
where
1
<i <
k.
::
w
(8.l).
To prove (8.2) we note that, if
there exists one and only member,
E' ,
is in A,...... E'.
Now the significance of the numbers
;:
is clear, as well as the truth of
P is a point of
P
w
2t-k < w < 2t-1.
Hence
Xi
-
E'
counting the mumber of points of
multiplying the results by
~
of..,,8 such that
in two ways and
q-l, we get
(qt-i+l _ l}X
i
i=l
E'
A,
=
q2t-k+l _ 1.
Adding (8.l) to this, and dividing the result by
t-k+l
q
we get
(8.2).
,
This completes the proof of Lemma 8.3.
Corollaries
1, 2
= 4,
follow readily from
Len~
8.3.
Similarly, if, in Lemma
t > 4, we get three cases, each as precise as the two cases of
8.3, we take
k
Corollary 2.
As is easily seen, the situation is much more complex if
t > k > 5.
= =
Now we are ready for
Example 8.4.
=====:::======
We begin by discussing a type of presentation
(8.3)
subject to axioms
(4), (5), (6) in which the projective plane
(q being a prime-power) and
(8.4)
-e
E
=
pG(6, q)
~ has order q3
44
For all we know to the contrary, the additional requirements which we impose on
the presentation need not forcenr to be even a translation plane, let alone
Desarguesian.
However, as a proof of consistency, we conclude by showing that the
Desarguesian plane of order
Since
C?,
has order
rr
q3 has such a presentation.
c:? 1? or of ~
the number of members of
is
given by
(8.5)
members of
members of
i.
eJ2
e
IcY2 t
=~v~v~
p
= 1,
l~ I = q3
1
(8.8)
~
=
ot4 v
(8.9)
1~4'
=
q+l,
~3
(8.10)
,
I(J I
+ q2,
6
=
q
- q
q3_ q , lci:3'
=
q
0
2
=
6'1
1
c?
1
0
o
q+l
6
i-dimensional members of
,
~
o
c?
q3_ q
Equations (8.6) - (8.10) are to be interpreted as follows:
collection of all
,
v£':3
let3'
0:'2
--
=
ep
(8.6)
(8.7)
Ie I
=
Ep '
,;e4
and
(Pi
denotes the
at 3 v£.:3
denote,
respectively, the collection of all 4-dimensional and all 3-dimensional members
45
of
e 2'
£
The collections
and £
3
:3
are disjoint and can be distinguished
according to table (8.10).
Table (8.10) is to be interpreted as showing the
number of members of
of each of the three types contained in a member of
e2
q+l
L.>0
p
of any specified type.
ep
lines in
and
For example, if S
q3_ q points in
disjoint, the number of points of S
(q+l)(q+l)
as it should be.
type and
cp'
and if S
contained in the members of
(:
P
S
contains
epare
is
(~_q) = q3 + q2 + q + 1 ,
+
Na;bu.rally, (8.7)
If S, T are distinct
then
Since the members of
!l~t;mbers
of
On t:re other haud, if
is the member of
e2
(8.11)
S
(8.12)
dim S
ep
gives the number of members of
gives the analogous numbers for ~
(8.8)
member of
t:'p'
'~3
is in
>
==
S
~
A, Bare dist.inct members of
containing A, B,
::>
.
then, by axiom (4),
GI'
A+B
of each
T 1s a
ep
we have
,
dim A +
dim B + 1.
There will be equality in (8.11) precisely when there is equality in (8.12).
particular, equality must hold in (8.11) if A
A
€
cP2'
B
€
0:>0'
determines the type of
£ 3'
that 1::
2
S e:
cP2'
B
(PI'
€
or
The reason is that, in each case, table (8.10) uniquely
S, and equality results in (8.12).
(8.11) if A, B e: 0-,:)1
holds in
€
In
and if
S
is not in
Similarly, equality
(~.~\;
However, this discussion makes it clear that axiom
in this case,
(4') fails and
has no rigid lines.
The above discussion allows us to deduce that a table showing the number of
members of
0_
e2
of each type which contain a member of
must have the following form:
~
p
of specified type
46
et 4
'P
:.:>(.
q
q+l
(?l
1
0
c?
(PO
0
1
q3
(8.14)
of
such that
containing A
o
then
0
r?-q
To verify (8.14) we consider a member,
8 ,
o
oC'3
3
ep
A , of
o
is disjoint from
A
o
T
is distinct fram
T
=
S,
8
and a suitably chosen member,
If
,
0
T
T "'"' S
o
o
is a membe!' of
Cit
e;
p
is a member of
and we will have
(8.15)
(T ~ 8 )
+
A
o
0
To begin with, let A be the
o
for every such T provided the dimensions match.
unique member of
8
joint from
of
q3_ q
and let
and T "" So
0
(?o
members of
8
is either one of
we are given the unique member
ep
of
~4
let
So
member,
r? -q
and tre
be a member of
T,
e.p
of
OJ1
opposite
6'2
tY2 )
.-
Ao
= 1,2,
Ai
q6
members of
r:l:. 4
0
£3'
2
of
..;e
is one of the
;
@o
3
and let
(8.14).
o
(J> 1 or one
It shows moreover that if
S ,
et 3
of
o
Next let A
o
disjoint from
0
1
is dis-
A
we can determine the
8,
which contains A.
8 , 8
Then
members of
6~, one member,
contained in
members of
be a member of
in table
of
A '
o
q+1
be a member of
Again we get
and
members
c?l
and
(8.15) for every
And this time we can verify the line
At the same time we note that if we are given
£
4 (so that
8
1
'"' 8
2
is the unique member of
~ contained in Sl or 8 , we may expre ss
together with all members of
each of the
i
A
o
in tables (8.14).
two distinct members
q+l
3'
We must conclude that (8.15) holds in either case.
This verifies the line opposite
all the members of
c1:
be any member of
0
uniquely in the form
members of
80
CPl
~
2
+ A
2
contained in
be a member of
..:e 4'
Where, for
8 ,
1
Finally, let
Since a member,
T,
47
of
et
which contains
A
o
hence the line opposite
must be in
in table
A:)
'-.To
we verify
;(3 or
(8.15) and
(8.14).
At this point it should be clear how we might attempt to construct a presentation
(8.3) with the properties explained above.
ever, whether even one such presentation exists.
It is by no means clear, howWe may obtain a (Desarguesian)
example as follows:
Let
(8.16)
E' I
1! (
for the case that F = GF(q)
be determined as in Example 8.1
(8.17)
E'
each member of
_
e,p
is a projective
=
projective plane (of order
E'.
q3)
Here
and each member of
e,t
ro(8, q)
is a projective subplane of
5-space of
and t = 3.
Moreover
over
E "
(8.16) represents the Desarguesian
GF(q3).
Our first step is to show that we
can find at least one 6-dimensiona1 projective subspace,
e t' .
contains no member of
e p'
distinct members of
then
S
= Al
+A
2
A , of
disjoint from S.
p
3
may choose a line L of
E' in Ai for
i
disjoint,
E
=
L + L
2
l
L + L +
2
l
tains a member of
0_
E' such that I
We may show this as follows:
e'
a member,
E , of
If· A , A , are
l
2
is a member of
and there exists
e£,
Since each A.
~
i = 1,2,3.
is a plane of
Then, since
L , L
1
2
E', we
are
is a 3-space of S, and, since
r.,
t::. £ t
has dimension
6.
S, L are disjoint, the space
3
If the 6-space
E, so constructed, con-
then (compare Example 8.2), for each member
either AcE or
A ,.... E is a point.
is a line, i = 1,2,3.
Hence
A of
Thi s contradicts the fact that
E contains no member of ~£t.
C3
p'
A ........ E
1.
48
Now let
E be any
no member of
6-dimensional projective subspace of
=
dim(A "'"' E)
dim A +
>
Hence A
#""\
e pt,
If A is a member of
E has dimension
2 +
0, 1
X
dim
6 - 8
or
the presentation
(8.16)
e p'
A + B,
If
4
of
S
~
eg',
Moreover, by Lemma 7. 2, the mapping
->
X ""'" E ,
upon a presentation
a contradiction.
e g"
is a member of
dim(S.- E)
Hence
dim S +
>
5 + 6
(8.3)
= PG(5, 9)
of
8.3
(8.3)
S of
replaced by
8.3
i
= 1, ••• , k,
Xi
E , of dimension
8
that
"-
t
= 3):
E'
ep
Hence
contains at most one member of
E
then
t
~
contains the
= 3.
Now we may apply the
t
= 3;
the
S; the spread...J of Lemma.
of Lemma
6-k.
(6).
(8.4) - (8.10).
dim E - dim (A + E )
is the number of members
dim (A ....... E ) = t - i = 3 - i.
and
A, B of
is replaced by
~J'; the subspace
S n
also satisfies
with the follo"Ting replacements:
Lemma
(4), (5)
subject to axioms
is replaced by the spread consisting of all members
member
induces an isomorphism of
3 or 4 for each S in C g t •
S /"\ E has dimension
Corollaries of Lemma.
E
E',
then
=
=
E)
= 0 .
2.
It E contains two distinct members
_
dim (A +
E -
We wish to show that the resulting presentation
member
which contains
then
X ranges over the projective subspaces of
where
E'
A of
8.3 ,
In particular,
A of
~~
eg'
contained in the
of dimension
k=2
8.3
or
eu.ch that
2t-k, is
3.
A c
Also, for
Sand
Thus we have three possibilities (When we remember
dim (Sr\ E ) = 4;
~ = 1,
x2 = q3 .
(II) dim (S ~ E ) = 3;
~ = 1,
x2 = 0, x3
(I)
= 3; ~
(III) dim (S/"'\E)
The cases
of
c:
contains a member
P, t
A of
contains at most one member of
member
of ~p,'
S
A,
o
p
~,
p
of~'
Then, for every member
A.
o
~ t
=1,
=q
+ 1,
X:; = c?
which is also contained in
- q.
E.
Since
e',
p
A of
Consequently, there most
A
0
/)
We choose
E is a line.
distinct from A, the member,
o
of ~I must obey case (I) or case (III). Of the q3 + 1 members of
o
p
containing A, let y obey case (I) and the rest obey case (III). Then, by
o
A + A,
counting the points of
E
S
, we deduce that there certainly exists a
such that
p
2
occur precisely when the member
which corresponds to case (III).
exist at least one member,
such an
~
(I), (II), corresponding to
= 0, x
= q3 .
ep,'
in two ways, we deduce that
(q+l) + y(q4 + q3 + q2 + q + 1 _ q _ 1)
+ (q3 + 1 _ y)( q3 + q2 + q + 1 _ q _ 1)
and hence that
y
Furthermore, if
cPi
In particular, E
denotes the collection of all spaces of dimension
e
£3 /
If now we take
ponding to cases (I), (II), (III) respectively.
unique member of
ep
t
contained in
E
e'p .
contains a (unique) member of
E, A E
t ,
equations (8.6), (8.7) hold. Next we define
to
p
be the collections of all spaces of form S /"'\ E, S t!!i
form A _
;e 3'
= 1.
A
o
i
and
;e4'
E.e',
corres-
to be the
and consider the members of
e£
con-
taining A, we may use elementary counting to deduce the first two equations of
o
0_
(8.9).
(8.10)
Example
The third then follows immediately from
corresponds to cases (I), (II), (III).
8.4.
(8.5), (8.8).
Moreover, table
This completes our discussion of
50
Example 8.1, 8.2, 8.4
subject to axioms
should suffice to show that among the presentations
(4), (5), (6) there can be broad structural differences, even if
we assume that the presentations yield Desarguesian planes.
the differences once more.
If a projective plane
Let us point out
has a presentation of the
~
•
type in ExaTlIple 8.1, there is a maximum. amount of symmetry.
All point-spaces have
the same dimension, all line-spaces have the same dimensions, and all of the
axioms
(4), (4'), (5), (6)
are valid.
~
In particular,
is Desarguesian.
has a presentation of the type in Example 8.2 there is still a
symmetry.
g~eat
If
~
deal of
Here one (ri3id) line-space plays a special role, yet all affine point-
sapces have the same dimension, all ideal point-space£,; have the same dimension,
and all affine line-spaces have the same dimension.
Axiom (4') fails but axioms
(4), (5), (6) hold and (since one line-space is rigid)
e l a t i o n plane.
is necessarily a trans-
Finally, we may regard Example 8.4 as representative of a much
broader class of presentations satisfying axioms
space is rigid.
~
(4), (5), (6).
Here no line-
Even more, every line-space contains point-spaces of distinct
dimensions and every point-space is contained in line-spaces of distinct dimensions.
We are at present largely ignorant of the nature of presentations of this latter
type, aside from the fact that they eXist, but surely
'\'le
the future a rich harvest of new combinatorial designs.
-e
have the right to expect in
51
~g~=g~~~~c;~~~~~g~~~~gg=g~~=~!~~~~~~gg~~~~~.
9.
We begin, for conven-
ience of reference, by repeating a construction, given in Bruck and Bose (1], of
an affine plane
1C
Here
= rc
( E ,
E',....g).
2t,
E is a (Desarguesian) projective space of (some) finite even dimDnsion
E'
is a hyperplane of
collection of
point of
E'.
E which are not contained in
t-dimensional subspaces of
in mexubers of
It was shown that
1C
E
..J.)
The points of
The lines of
Et.
which are not contained in
...g,
(Thus...J is a
E' having the property that each
is contained in one and only one member of
Et
Et
is a spread of
(t-l)-dimensional subspaces of
are the points of
sect
..g
E, and
Et
rc
1C
are the
end which inter-
one for each line.
is an (affine) translation plane.
Moreover, if
1C
is
some (affine) translation plane, and if R is one of the coordi,natizing right
Veblen-Wedderburn systems, a necessary and sufficient condition that
representation
field.
(9.1)
In addition, if R has finite dimension
E of dimension
In [1], the
2t
t
over some skew-field F
1C
has a presentation
(9.1)
over F.
·~aff;1l1!e plane
(9.1) was imbedded in a projective plane
Et,
--
have a
is that R be finite-dimensional over its left-operator skew-
contained in the left-operator skew-field of R, then
with
1C
,,&)
by adjoining the members of,<f to
1C
to
If, for our line at infinity, we use the hyper-
plane
(9.3)
1C
as the "line at infinity."
E' instead of the spread
1£ (
as "points at infinity" and the spread
1,
we get a presentation
E ,
~,
(?.e)
52
eJ,
(4), (5), (6) and such that
subject to axioms
r3.p
(Note that, in terms of (9.1), the members of
members of
..J;
and the members of
eJ,
has a rigid member, namely I:
are the points of
are the lines of
and the
1t
and the hyperplane
1t
I: t.)
The presentation (9.3) satisfies the hypotheses of Lemma 6.2 (as well as
axioms (5), (6»
and corresponds to the case
i
= t,
a
= 1.
Since we can readily
go back from (9.3) to (9.1), we now have the complete story (in so far as existence
is concerned) about presentations (9.3) which satisfy axioms
have
i
= t,
a
Either
(4), (5), (6) and
= 1.
(9.1) or (9.3)
translation plane.
can be considered as the basic representation of a
In the sections which follow, we shall favor
(9.1).
Before we leave this topic, we wish to pursue Lemma 6.2 a little farther.
~
Suppose that R is a right Veblen-Wedderburn system having finite dimension t
over some skew-field K contained in the left-operator skew-field of R.
further that K has finite dimension,
of K.
If
presentation (9.3)
member of
dimension t
or 2t-1 over K.
we can represent I: as a projective spa.ce
(over F).
~
= st,
a
= s.
I:
*
over F in
(sk - l)-dimensiona1 projective subspace of
(9.1) by a presentation
~,~;,
which satisfies the hypotheses of Lemma 6.2
i
~.e has
1, a (k-1)-d1mensiona1 projective sUbspace
In this way we replace
1t (
has
has a
More or less as in Example 8.1 (but with 3
of I: (over K) is represented as an
*
1t
where E is a projective space of dimension 2t over K, each
such a way that for each integer k
I:
then
E'p has dimension 0 or t-l over K, and each member of
replaced by s)
•
s, over some subfie1d F of the center
is a projective plane with coordinate ring R,
1t
Suppose
eJ,
*)
(and also the axioms
(5), (6» and
From such a representation, if s > 1, we can clearly get
I
more complex representations, analogous to that in Example 8.4.
I
For the rest of this paper we shall be occupied with (9.1).
t
53
10.
~g;;ea~~=~~=~;~~=g~_~;;~~fQ~~~~£~~. In
dence between spreads and sets of matrices.
[1]
we described a correspon-
Here we rephrase matters slightly,
replacing matrices by linear transformations.
Let F be a skew-field,
t
~
2 be a positive integer,
(2t - l)-dimensional projective space over F.
=
(10.1)
of
£,
£'
£'
be a
in form
£ (V/F)
where V is a vector space of dimension
Then a spread,-d(,
We represent
and
over F as a ring of left operators.
2t
becomes a collection of t-dimensional vector subspaces
of V over F having the property that each nonzero vector of V lies in
precisely one member of
~
.
Let A, B, C be an ordered triple of three distinct t-dimensional vector
subspaces of V, mutually disjoint in the sense that each two have only the
zero-vector in common.
Then there exist one and only linear transformation
a -+ a'
of A upon B over F
such that the linear transformation
a -+ a + a'
maps A upon C.
To each linear trans formation X of A into A over F there
corresponds a unique t-dimensional subspace,
(10.2)
J(X)
=
J(X), of V given by
{ a X + a'
a
E
A}.
Then we define the first of
(10.3)
J(oo)
=
J(O)
= B
J(I) =
--
A =
C
( a
a
€
A)
a
€
A)
-
(a'
=
(a + a'
The last two definitiona are special cases of
a
A)
(10.2).
understand that
A, B, C are mutually disjoint
of V and that
a -+ a'
discussion so far.
€
However, as long as we
t-dimensional (vector) subspaces
is a linear transformation, (10.3), (10.2) sum up our
We should add that each t-subspace
J
of V which is dis-
54
joint from A = J(oo)
has form (10.2) for a unique linear transformation X of A
into A.
If X is an linear transformation of A into A over F, let us define
(10.3)
null X,
the nullity of X, to be the dimension of the subspace of V consisting of all a
in A such that
(10.4)
aX =
O.
Then, obviously,
vector dim (J(X),.-'""\ J(Y»
In particular,
J(X), J(Y)
=
null (X-Y) .
are disjoint if and only if X-Y is nonsingular.
Now let A, B, C be an ordered triple of three distinct members of a
spread, ~,
of ~ t.
Then, in terms of the above representation, ~ corresponds
uniquely to a collection
e
e
(10.5)
=
e(A, B, C)
of linear transformations of the vector space A (into itself) over F subject to
the following conditions:
~
(i)
contains 0 and I.
If X, Yare distinct members of ~ , then X-Y is
(ii)
nonsingular.
(iii)
in
e
such that aX
If a, b E A,
=b
a 1= 0, there exists one and only one X
.
Note that (iii) states that the nonzero vector b + a t
unique member,
is contained in a
J(X), of ~ •
We note that conditions
(i), (ii), (iii) are concerned with a collection
of linear transformations of a fixed t-dimensional vector space A over F.
-e
c: 1
is a collection, with properties
(1), (ii), (iii), of at-space Al
If
over
55
F, and if P is any one-to-one linear transformation of A
l
then the conjugate set
=
(10.6)
P
-1
>0
I...,..
P ,
1
p-lxz
consisting of the linear transformations
upon A over F,
P,
Xl
in
~l'
has
the properties
(i), (ii), (iii) and involves A rather than A • We shall make careful use of
l
the process of conjugation in discussing the following operations on a collection
(I)
Inversion.
X
-+ X* , for all
X
in
C', where 0*
=0
and x*
= X- l
if X ~ 0 •
(n) Affine transformations.
of
Q, R
(10.7)
For two distinct but arbitrarily chosen members
18,
=
X -+ X*
for all X in
(R _ Q) -1 (X _ Q)
t:::. •
Let us begin by noting that if
~ has properties (ii), (iii) but not
necessarily (i), then the transformation (10.7) maps ~ into a collection
with all of the properties (i), (ii), (iii).
It will be convenient to write
(10.5) in the form
(10.8)
~(J(~), J(O),
=
J(I».
To understand inversion, let us attempt to replace ~ by
C* =
~ (J(O), J(~), J(I».
If we define the linear transformation
a e
--
then we note that
J(~)e
=
if
a'
,
e
a' e
of V by
=a
,
'V a G A =
J(~)
is the identity transformation of V, that
= J(O),
J(o)e
=
J(~)
e *
56
and that, for
e,
X E
X~ 0 ,
J(X) e
This means that
e
e
=
( aXe + a'e
=
( a + (aX) I
=
( aX- l + a'
=
J(X- l )
Ia
Ia
Ia
E
J(Q)}
E
J(Q)}
E
J(Q)]
is obtained from C; by inversion.
* e
To understand the
affine transformation (10.7), we attempt to replace ~ by
t: * =
J(Q), J(R) ).
e(J(Q),
Here we attempt to define a linear transformation
~
of V by the requirements
tmt
=
a cI>
a,
(a +
for all
a
in A
a'el>
= a KQ +
(a K)'
al)~
=aLR
+
(aL)'
= J(Q),
where
,
K, L are linear transformation of A into A.
Then we must have
I
+
KQ=LR,
K
=
L
K=L ,
whence
Therefore, for every
=
X in
(R _ Q)-l .
~,
if b
=a
(aX* + a')' = a(X* + KQ) +
provided that
x*
J(Q)41
for all
X in
is given by (10.7).
whence
up these results in a lemma:
=
41 ~* 41-
,
(a K)'
= bX
+ bf
Now we have
= J(Q),
e,
K
1
J(X),
consists of the matrices
X*.
We may sum
•
57
~;~=~~=~. _1J_et
~_'_b_e_a--,( ..2,-t_-_1...
) -_di_·_m_e_n_s_i_on_a_l_p...r...o~j_e_ct_J._·v_e_s.....p_a_c_e_o_v_e_r_a_sk_e_w_-
field F, expressed in the vector form (10.1').
Let A, B, C and AI' B , Cl be
l
two ordered triples of mutually dis.joint members of a spread .£ of
E'. Then, if
e
=
(:?(A, B, C),
there exists a collection
~
*
obtained from
e. by
iteration of affine trans-
formations and inversion such that
C'l
where
maps
P
p-l
c:. *
P
is a nonsingular linear transformation of V upon
AI' B , C
l
l
Proof:
~
=
upon A, B,
V over F
which
C respectively.
In view of tre preceding discussion it is enough to note the
following schema, which indicates the effect of successive transformations:
x
Xl
X
3
=
(S _ R)-l (X _ R)
X
= X-I
2
1
(T - I)-1(X - I)
2
2
Here it is assumed that
of .~.
00
R
S
T
00
0
I
T
l
0
00
I
T
2
00
0
I
J(R), J(S), J(T)
are three mutually disjoint members
(In the last line of the schema, the dash represents a linear trans-
formation in which we are not interested.)
58
~~g~:~;~;=~~g~~;~~~~=g!=~g~~~~~.
11.
terminology in connection with spreads.
It will be convenient to introduce some
If (as in section 10),
of a (2t-1)-dimensiona1 projective space
E'
Each collection
~
or as a t-spread over a
defined as in section 10 (See, for example,
(10.5» will be called a representatio!! of
In addition, if
is a spread
over a skew-field F, we shall
speak of~ as a t-spread with coordinate skew-field F
skew-field F.
~
,g (by
linear transformations.)
is an affine translation plane defined by (9.1), we shall
1[
call the defining spread ~ a Moufang, Desarguesian or Pappus spread according as
the plane
1[
is Moufang, Desarguesian or Pappus.
Let us recall here that an
affine or projective space is a Pappus space if and only if it satisfies the axiom
of Pappus or, equivalently, if and only its coordinate rings are (isomorphic)
fields.
AlSO, a projective plane is Moufang if and only if it satisfies the Little
Desargues l Theorem; an affine plane is a Moufang plane if and only if the corresponding projective plane is Moufang.
And Moufang planes are those Whose coordinate
rings are (isomorphic) alternative division rings.
The discussion in [1]
regarding the coordinate rings of a translation plane
may be rephrased as follows:
(9.1), Where, for t ~ 2,
a representation of
A over F
~
Let
,..g
1[
be an affine translation plane given by
is a t-spread over a skew-field F.
Let ~ be
by linear transformations of a t-dimensiona1 vector space
(as a ring of left-operators.)
Choose some nonzero vector, 1, of A
and <lefine multiplication in A by
•
a(lX) = aX
(11.1)
for every a
in
A
and
X
in
e.
Then the system
notes vector addition, is a coordinate ring of
Wedderburn system.
_
affine plane
in thi s manner.
1[
1[
,
(A, + ,.), Where
+
de-
namely a right Veb1en-
Moreover (to within isomorphism) every coordinate ring of the
(though not of the corresponding projective plane) may be obtained
59
Although the properties of the right V. W. system (A, + , .)
will depend
to some extent upon the choice of the identity element 1, we shall consider
only properties for Which the choice of 1
is
irrelevant.
We begin by stating
the more immediate properties in an informal manner:
(A, +, .)
(11.2)
is a division ring precisely when
addition (and then
~
(I:, +) is an a.belian group isomorphic to (A, +).)
(A, +, .) is a (right) near-field precisely when
(11. 3)
multiplication (and then ( t;, .)
(A, .)
(11.4)
~
is closed under
is a group with zero, isomorphic to
.1
(A, +, .)
is a skew-field precisely when
t:lel't' (e
,
(¢'. +, .)
is a -r.~ .(anct-
+" .,) ll.s a sl~ew-field isomo;ryhic to (A, +, .).)
We shall treat these properties in turn.
are uniquely defined elements
If X, Yare elements of
~
, there
S, P of (:;. such that
leX + -i) = 16·
(11.5)
is closed under
,
.ntY~ ~
...l P« -••
Let us define
b =
Then, for all
a
.lX·,
c
=
-1I.
=
a(X + Y)
in A,
ab + ac
=
aX + aY
a(b+c)
=
a.( 1X + U~)
"
= a(1S) = as
Hence we will have the left distributive law
ab + ac = a(b+c)
for all
a, b, c
in A if and only
is, if and only if C
the mapping
S = X + Y for each choice of X, Y;
is closed under addition.
When C
that
is closed under addition
60
x
(11. 6)
(C::
is an isomorphism of
a right
V. W.
Again, with
, +)
~
1X
(A, +).
upon
Since a division ring is merely
system with the left distributive law, we have proved (11.5).
8,
b, c
as before,
(ab)c
=
(aX)c
=
aXY ,
a(bc)
=
a(cY)
=
a(lXY)
a(lP)
=
aP.
=
Hence we will have the associative law of multiplication if and only if
for each choice of X, Y; that is, if and only if
When C
(~
e
(A, .).
V. W.
is an isomorphism of
system (A, +, .)
is a right near-
(A, .) is associative and hence a group with zero, we have
field if and only if
proved (11.3).
Since a right
= XY
is closed under multiplication.
is closed under multiplication, the mapping (11.6)
, .) upon
P
Since a skew-field is merely a division ring with associative
multiplication, (11.4)
is cow obvious.
Our first theorem depends for its truth
upon deep known results:
Theorem
11.1.
============
let
e
For
t
> 2, let A be a t-spread over a skew-field F, and
be a representation of..8 by linear transformations.
Then each of the
following statements implies all the others:
A8
(i)
(ii)
is Moufang .
Every representation of.J by linear transformations is
closed under addition
.
e J.s
(iii)
nonzero
X
in
~
(iv)
closed under add! tion and contains
X- 1 for every
.
~
X, Y in l:': with Y 1=
is closed under addition and contains
o.
XY-lx
for all
61
~~~~~~~~on=~t~~g~!. (For various statements which we leave unproved, and for
adequate references to the literature, see Bruck
[5], Hall [3].) (i) is equiva-
lent to the statement that every coordina.te ring of
plane (9.1»
(the affine translation
1t
is a division ring and hence (by (11.2»
is equivalent to (ii).
Next we shall show in deta.il that (ii) implies (iv).
, +)
By (ii) and (11.2), (c.
( e *,
+ ) is an abelian group if
Yare in
Hence we
<= *
):, with Y ~ 0, then
assume
elements of 1::.
is an abelian group. By this and Lemma 10.1,
~
*
e
X y- l X is certainly in
X ~ 0, Y and set
Hence
is obtained from
Z = Y - X.
Then
by inversion.
If X,
~ if X = 0 or Y.
X, Z and Y are nonzero
contains
We note tr,at
X WZ
Hence
=
W is a nonzero element of
-1
W
X-W
Thus (ii)
implies
(iv).
~*.
~
Therefore
contains
-L_
-1
=ZYJC=X-XY X
-1
and
W= X- l Y Z-l
Z + X = Y,
=
XY
-1
X.
Taking X = I, we see also that (iv) implies (iii).
Next we consider the meaning of (iii).
First let us assume merely that
(11.2), (A, +, .)
of A and let
(11.7)
b
is a division ring.
b-1
= 1X
is closed under addition so that, by
Let
b
be an arbitrary nonzero element
be the unique element of A such that
,
,
for unique (nonzero) elements
_
e
X, X'
1 X X'
of
e.
b b-1 = 1.
Then
=1
Moreove; if X is an arbitra.ry
nonzero element of C , there exists one and only one element
X'
of
<::
such
62
that (ll.7) holds with b = lX.
{ab)b
•
Ml
=
{ab)b- l
a, b
in A
=
with
a
in A,
ax X'
(A, +, .)
Hence the ditision ring
(for all
For all
will have the right-inverse property
a
b 1= 0) if and only if
(iii) holds.
However (and
this is a deep result) a division ring has the right inverse property if and only
if it is alternative.
This ends our sketch of the proof of Theorem 11.1.
For the sake of completeness, we restate (11.4) as follows:
~g~g~~ ~~=~.
Under the hypotheses of Theorem 11.1, a necessary and sufficient
condition that.A8 be Desarguesian is that
Proof:
-====
(e.,
+, .)
be a Skew-field.
Obvious from (11. 4) and Theorem 11.1.
We may remark that Theorem 11. 2
leaves something to be desired:
We should
like more precise information about the relation of the skew-field (e , +, .)
to the coordinate skew-field F
of..J7.
Our next theorem gives sharper results
for Pappus spreads.
~g~~:~~=~~~~. For
t ~ 2, let
.-<!
be a t-spread over a skew-field F
let C be a representation of A! by linear transformations.
and
Then we may dra.w the
following conclusions:
(i)
(ii)
{~
If ~ is a Pappus spread, then F
A necessary and sufficient condition that
,+,.)
(iii)
is a field.
If
-.J
be a Pappus spread is that
be a field.
{e,
algebra over F.
is a field, then F
+, .)
More precisell,
e
t-l
1:
i=o
c
i
(e,
+, .)
is an
contains an irreducible linear transformation
X and consists of all elements of form
where the coefficients
is a field and
lie in the field F.
63
....g
~gg~ .
(A, +, .)
•
is Pappus if and only if some (and hence every) coordinate ring
of the corresponding translation plane ..,.., is a field and hence if and
e ,
only if (
+ , • )
As shown in [lJ, if A = (A, +, .)
is a field.
coordinate ring of the translation plane (9.1), then F
left-operator skew-field of A.
field.
f
Next suppose that
be in F
aT
e
That is,
in F
T
and
= a(l
is a field, so that
=
(~,
T)
= a(f
1)
= (fl)a =
i(la)
e,
= fa
f I.
Consequently, for every
t; contains the transformation
+ , .)
(f I)X = f X.
is an algebra (of dimension
(9.1), we pick a line
and a projective subplane,
then
E
2
E/
L of
~,of
E'. It is easy to see that, if
E
2
Let
.
t)
over F.
*
is a field.
1
The rest of
Starting from
~,
but is not contained in
is the extension of
is isomorphic to a subplane of
is Pappus and F
Now it is
which is contained in no member of ~ ,
E which contains
~
f
This completes the proof of Theorem. 11. 3.
We note, in passing, the following geometric proof of (i).
plane, then
is a field.
F
in A, by commutativity ,
the conclusion of (iii) should be clear.
given by
is a
fl.
is the scalar linear transformation
X in
clear that
In particular, if A is a field, then F
( ~ , + , .)
lT
a
is a sub-skew-field of the
T be the l.mique element of l::.. such that
and let
Then, for every
is a
#.
~
to a projective
Hence, if
~
is Pappus,
64
12.
Before we introduce the concept of regularity we wish
~~~1~~12r~~.
---------=--
to review some well-known facts.
be a projective space and let X be
E'
Let
E'.
a non-empty collection of non-empty, mutually disjoint projective subspaces of
Then by a transversal to
~.
we mean a line,
point in common with each member of %.
L, of
E' which has exactly one
In particular, if
~,
for two disjoint, finite-dimensional projective subspaces
then it is easy to see that each point of
E + E2
l
E' =
E 2 of
E', other than a point of
E 2' lies on exactly one transversal to
E')
E 1 or
~.
Now let us specialize to the case that, for
t > 2,
=
A, B, C are three
mutually disjoint (t-l)-dimensional projective 8ubspaces of a (2t-l)-dimensiona1
E'.
projective space
e
Then each point of A,
transversal to A, B, C;
B or
C lies on exactly one
and two distinct transversals are skew (i.e., disjoint.)
Let L be a transversal to A, B, C.
We wish to determine the conditions ensuring
that to every point P of L there corresponds at least one
projective space D containing P
exactly one point in common with D.
to A, B,
~
C distinct from L.
=L + M
r.,;
B~ C~
K,
D in a line
For this purpose, let M be a transversal
Since
L, M are skew, the projective subspace
Every transversal to A, B, C is either disjoint from
contained in
skew lines
such tha.t every transversal to A, B, C has
3-dimensiona.l and meets A, B, C in the lines A', B', C'
is
respectively.
(t-l)-dimensional
D'
to A', Bt, C'
in the latter case, it is a transversal in
through P wi th the property that every transversal in
is also a transversal to D'.
~
transversal in
r.,
of
to A', B', C'.
~
~
meets
E3
Hence, if D exists for every P,
is a Pappus space, then to each point
ponds one and only one line D'
or
to the three
If D exists for a given point P of L, then
a well-known property of projective 3-spaces tells us that
Moreover, if
~
-S
which contains
-S
is a Pappus space.
P of L, there corresP and meets every
In particular, for the property which we want,
65
E' and
we must assume that the coordinate skew-field of
~
Let us specialize onceasato. For t
•
E'
2, let
E 3 is a field.
be a (2t-l)-dimensional
projective space over a field F, and let A, B, C be an ordered triple of mutually
disjoint (t-l)-dimensional projective subspaces of
form (10.1) and A, B, C in form (10.3).
vector spaces
(a) ,
taining the point
tains the point
F,
E A, a ~ O.
a
{a}
is the line
We represent
E' in
The points of A are the l-dimensional
The unique transversal to A, B, C con-
(a, a')
.
We observe that A = J(au) con-
(a) for each a ~ 0 in A and that, if k is an element of
J(kI) contains the point
lka +
incidentally, that the mapping
kI
a(kI)
for every a
E'.
in A.
Hence
kI
a~
for each a
~
0 in A.
.. We must note,
is defined by
=
lta
is a linear transformation of A for every k in
F only because the skeW-field F
is in fact a field.
-
Since, in the preceding
paragraph, we saw that there could be at most one D for each point P of L, we
now see that the spaces A
projective spaces of
C
= J(I)
= J(oo)
and J(kI),
kEF, are all the (t-l)-d.imensional
which meet every transversal to A = J( 00), B = J(O),
E'
in exactly one point.
The collection consisting of these spaces we call
the !egulus
•
a
(12.1)
=
a (A,
corresponding to (or containing)
tive space
E'
B, c)
A, B, C.
over a field F, (t
ThUS, for a (2t-l)-dimensional projec-
> 2) a regulus is a collection,
mutually disjoint (t-l)-dimensional projective subspaces of
properties:
e
(i) OG has at least three distinct members.
to three distinct members of
a
is a transversal to
cR..
a,
of
E' with the following
(ii) E'\,B ry transversal
(iii)
transversal to (}2, each point of L lies on a (unique) member of
If L is a
Ci£ .
66
In the case
t
= 2,
the members of
a
are lines.
versals to A, B, C (and hence to the members of
•
a',
regulus
the opposite regulus to
symmetrically related.
For
t
a,
In this case, the trans-
a ) themselves constitute a
and the two reguli
For
t ~ 2, let A! be a
three distinct (and hence mutually disjoint) members A, B, C of
cR (A,
e
-<!, A
CRJ.A, B, C) has only the three members A,
That is:
=2
or
every spread over GF(2) is regular. It is known that, if
3, all planes of order 2t are Desarguesian and (hence) Pappus. We
shall be unable to give any further information in this paper about t-spl'eads over
GF(2)
where
t
is a prime.
fields other than GF(2)
~g~~:::~=~&=~.
For
As the following theorem shows, regular spreads over
are very well-behaved:
t
~
2, let F
and let ..,.g be a t-spread over F.
be a field with at least three elements
If...,g ia a Pappus spre ad, then ,.,J is regular.
~ is regular, then ~ is a Moufang; spread.
If
~~::g~~m.
If t;: 2
and if
over the finite field GFeg)
Proof.
=====
q
>2
where
q is a prime-power, at-spread
is regular precisely when it is Pappus.
The Corollary depends on the known fact that, for finite projective
planes, the Moufang, Desargues and Pappus properties are all equivalent.
•
con-
B, C).
We note that if F = GF(2), then
t
are
We define A:f to be regular provided that, for every
t-spread over a field F.
B, C.
CR. '
> 2, the situation is clearly different.
Now we come to the concept of a regular spread.
tains the regulUS
a,
We proceed
to a proof of the Theorem.
To begin with, let the spread
11, each representation
k
€
F.
e
We start from some
x -+ X*
=
..-J
be regular.
Then, in the notation of section
of ,J contains all the scalar transformations
e
and perform the affine transformation
(T - Sr
l
(X - s)
kI,
67
where
t:. *
must contain X* = kI
•
x =
for each
k
in F.
multiplication.
k
t:.
S, T are two distinct elements of
and
l-k
(1 - k)S
Taking
Since F
are nonzero.
for each
+
e
Thus
e
must contain
k T
e
must be closed under scalar
has at least three elements, we can choose
k
so that
Then, if W is defined by
k,
must contain
(1 - k)(W S) + k T
and hence must contain S + T.
e
in F.
S = 0, we see that
(1 - k)W =
we see that
k
The corresponding representation
= k(S
+ T)
Here we have assumed S 1= T.
e
However, since
is closed under scalar multiplication, we may now conclude that e' is closed under
addition.
Therefore every representation of
sequently, by Theorem 11. l,...g
...J
is closed under addition.
is Moufang.
If, on the other hand, we assume that ~ is Moufang (and that
then
1
F
is a field)
will be regular precisely when, for each representation ~ of A!,
is closed under scalar multiplication.
fact,
Con-
e
We have been unable to decide whether, in
is closed under scalar multiplication.
is true by (iii) of Theorem 11. 3.
e
How'ever, if
The proof of Theorem 12.1
,.g
is
Pappus, this
is now complete.
It seems a reasonable conjecture that regularity and the Moufang property are
equivalent properties for spreads over a field F.
is the field of rationals.
There remain
The conjecture is true when F
the field F = GF(2)
and the infinite
fields other than the field of rationals.
Before we leave this topic, we wish to record a simple but useful property
of regular spreads:
68
-
_._._.
__ __._-_. __
Lemma 12.2. For t ?
E' be a (2t-l)-dimensional
projective space
- 2, let ._-_.
.. _----_._==========
over a field F and let ..J be a regular spread of
E' . Let L be a line of
E , contained in no member of
•
4
and let
members of ~ containing points of
course,
be the collection consisting of all
There is no loss of generality in assuming that F
elements and hence that the line
taining relation of
L has at least four points.
Also,
L
3
Clearly the con-
a.
is a transversal to
t£, the regulus
three distinct members of
If A, B, Care
O?l = cR(A, B, C) has one and only
one member which contains a specified point of L.
O?1
has at least
E' sets up a one-to-one correspondence between the members
of (R. and the points of L.
Consequently,
E' (and, of
Then ell is a regulus of
L.
OG.)
L is a transversal to
;roOf.
02
By regularity, (}(1
is
in
"R,
(J?, as claimed.
=
E' be a (2t-l)-dimensional projective
space over a field F.
By a partial spread,
~,
of
E'
we mean a non-empty
collection, ~, of mutually disjoint (t-l)-dimensional projective subspaces of
By a proper partial spread, ~-G, of
which is not a spread of
E'
we mean a partial spread,
,of
E'
E' .
?G,
By a switching set,
of
E' we mean a proper partial spread, fj{, of
for which there exists at least one conjugate partial spread,
•
'?1::
E'.
%
I ,
E'
with the
following properties:
and %'
(ii)
%
and
%
I
have no conmon members.
cover the same points.
By (ii) we mean that a point
lies in a member of %
(i1), and hence that
I.
?G'
P
of
Note that
E' lies in a member of ~ if and only if P
?t'
is a proper partial spread, in view of
is a switching set with ?C as one of its conjugates.
Switching sets may be used as follows:
..
contains a swltching set ~.
-J' ,
collection,
of
x!
obtained from
Note that
E' •
"J
~,
and
~t'
Let
-8,
Suppose that a spread,
be any conjugate of
:x..
of
E'
Then the
by replacing ~ by 0-(.', is also a spread
are distinct but have at least one conmon
member.
--!, .c:f'
Conversely, let
least one common member.
J
Let
which are not members of
manner with
A
and
be two distinct spreads of
~
E'
which have at
be the collection consisting of all members of
,J' , ,
and let
'%, be defined in the same
.4' interchanged. Then:Jt.. and
~,
are conjugate switching
sets.
In the special case
Z'
space
Then
~
t = 2, let :?J<' be any proper partial spread of the 3-
which is a disjoint union of (one or more) reguli,
is a switching set with a conjugate,
07, '.
corresponding opposite reguli,
.?t; "
(R, of
E'.
consisting of the union of the
ThUS, in terms of spreads, a fairly rich
theory of switching sets for the case
t=2
has been sketched in [1].
We might
add that DaJ.e Mesner has found a switching set of a finite 3-space which is not a
disjoint union of reguli.
Returning to the general case
switching sets of
~, •
~
t
We may call
2, consider a pair,
:.x,
dim (J ......, J')
..
for a fixed integer
tively.
d
as
Here, necessarily,
J,
d
~
I
X , ?r.:,)
of conjugate
"C' a regular pair if
=
d - 1
J'
range over the members of ~,
1,
in view of property (ii).
obvious example of a regular pair is a pair of opposite reguli.
u..'
respec-
When t = 2, the
As the next
theorem shows, there exist many examples of regular pairs of conjugate switching
4It
sets for
t
> 2.
70
Theorem 13.1.
===========
(13.1)
where
Let
=d
t
d, s
Q = q
s ,
are positive integers,
(13.2)
Q>2.
=
such that, for each
(a)
=
{ A, B} \..;
PG(2t-l, q)
Pt =
--.J
~i
• ••
...."
~ Q-l
is a switching set of
with the following properties:
I?C I
A conjugate,
%
:?-G'
~t:,
for every J
Remarks.
======
f,
=
(1)
and
If
Jf
q
(13.3)
in
/"'I
J')
=
different
d - 1
~-{,'.
> 2, we need only assume that
Theorem 11.1 and Corollary.)
the decomposition
~( s )
of % may be chosen in at least
dim (J
in %
t
(q - l)/(Q - 1)
is a regular pair with
(13.5)
•
'-' ' i 2
The number of members of ?C is
ways so that
..
9t 1
i, the collection
(13.4)
(b)
GFeg), and if A, B are any; two distinct mem-
.;-s:S, then ~ ma;y: be expressed as a disjoint union
(13.3)
I;'
is a prime-power, and
~ 2,
s
If ,g is a Pappus t-spread over
bers of
q
d
(2)
For tJ:e case that
t
~ is regular.
= s = 2,
uniquely by requiring that for each
(See
we may characterize
i, the lines
be conjugate non-secants of the doubly-ruled quadric defined by the regulus
A, B
~i'
It is not clear whether an equally geometric characterization exists in other cases.
(3) As the proof will show, the case that
In connection with
s
(b) we actually construct
is a prime is especially interesting.
1 +
~(s)
=s
switching sets, one
71
of which is ~, such that every two of the switching sets form a regular conjugate pair with property (13. 5) .
Proof.
Let A, B, C be an ordered triple of distinct members of
====
we shall construct, in a unique manner, the switching set
(13.3) which contains C.
(A,
+, .) by
(11.1).
representation,
e,
We represent
..,g by linear transformations of
right multiplication R(a),
(13.6)
in A.
for every b
=
a
then ell
in the decomposition
The corresponding
A is the set of all
Here, of course,
ba
If we define the mapping 1> of the field A by
(13.7)
order
a e A, of A.
b R(a)
Then
A, B, C in form (10.3) and define
(A, +, .) is a field A = GF(qt).
Then
of
~
,.g.
~
= a
Q
,
"i/
aeA,
is both a linear transformation of A and an automorphism of A of
s.
The multiplication cyclic group generated by 41
generators.
Let
elements of A
Each
a
e
be anyone of these
fixed by
in A
=
e
(or by
GF(QS)
~
~
(s)
) form
~
has
generators.
distinct
Note that the
a subfield GF(Q)
has a relative norm,
(s)
= GF(QS).
of A
N(a), defined by
s-l
(13. 8 )
N(a)
=
n
a
i=o
Let ~/Z. be the multiplicative group consisting of the linear transformation
•
where
and for
a
has relative norm 1.
For <X we take the collection of spaces
X' we take the collection of spaces J( e "'m
are arbitrary nonzero elements of A and if
x(R(a) -
e R(b» =
=bc,
a
0
).
Note that if
then, for
x
in
A,
R(a)
I
a,
J(
(..m );
b
72
if and only if
(13.
g
9
It is easy to see that (13.
•
If
e =
x
xc.
has a nonzero solution
x
if and only if N(c)
o
(13.9) has a solution xo ~ 0, then the set of all solutions of
= l.
(13.9) is the
set of all elements of form
k X
o
where
GF(Q}
k
ranges over the solutions of
= GF( tId}.
(13.10)
null (R (a) -
according as
N(a}
(13.~)
(for
k; that is, over the elements of
In this case, the solutions of (13.
subspace of A over GF(q).
e
k e :::
vector
=
or N(a) ~
(J(R(a)
dim
a, b ~ 0) according as
~
J(
By this and (10.4)
=
R(b» }
d
or
~
0 ,
=
,
0
or N(a) ~ N(b).
and~'
a, b
0
N(b).
= N(b)
N(a)
% = J('1rl.)
enough to show that
form a d-dimensional vector
Thus, in the notation of (10.3), if
e R(b» = d or
N(b)
9>
J(em )
This is more than
form a regular pair
of conjugate switching sets with
dime ::T "'"' .J I)
Moreover, % is contained in
•
•
Note that if
chosen as
mod
~i
s
=
Null(
N(b) ~
rJ
R(a)
=C
.
(13.7),
Moreover, if 0, i, j
e
can be
are incongruent
0, then
-
~ R(b» = d.
s::: 1 + '(s)
0 < n < s, form a regular pair ;;1( , 0-t:' '
(13.12)
J(I)
4l is defined by
i ~ 0 mod s.
This means that every two of the
_
d - 1 .
and contains
is a prime, and if
for any
s, and if N(a)
..2
=
The corresponding situation when
sets
J( 8 n9Jl,),
of conjugate switching sets subject to
s
is not a prime is equally clear but
not worth stating.
It should be obvious that the
Q - 1 switching sets in (13.3) corresponds to
e
73
the
Q,-l cosets of' 7?(in. tnc' multiplicative 'group
granted the fact that the order of
~I
•
•
7/f. ,
(q
However, the multiplicative group of
a
of A has relative norm 1
r
/1(
Hence
:L1as order
r,
e
.=....
Here. we arc takinG fcor:'
and hence tne order of
t
15<1
=
or
J(
is
J
- lD/(Q - 1)
is cyclic of order
if and only if
ar
=1
qt -1;
and an element
,mer e
= 1 + Q + Q,2 + ••• + Qd-l
= (Q,d_l)/(Q_l) = (qt_l)/(Q,_l)
as we assumed.
This completes the proof of Theorem 13.1.
With very little change, we can
prove somewnat more, as follows:
Suppose as before that
t
= ds,
spread over an infinite field F
field A
= (A,
of order t.
+, .)
s
> 2, but now assume t h a t l is a Pappus
"tnich is a cyclic spread in the sense that the
is a cyclic extensman of F, with a generating automorphism
Then, as before,
A has an automorphism
point onwards, we proceed as before.
of order
~
The main difference is
s.
From this
that~'must now be
decomposed in terms of infinitely many switching sets.
Referring to the hypotneses and. proof of Theorem 13.1; suppose that
represented by ~
·e
I
..
, consisting of the matrices
in terms of
1
e
by replacing?J1. by
R(a), a
€
ell( ) then
andnence does not, properly speaking, represent a spread.
choose
a
)!(R(a)
in A so that
by
N(a) ~ 0; 1, and define
(1)( R(a), then
as is easily seen,
{)2
e. 2 from
Pappus (and hence, of course, not Moufang.)
is
If we define
A.
1
c:... 1 does not contain
However
if we
e by replacing
e 2 does not represent a spread
is not closed under addition.
£!
/'.
Moreover
Therefore / . is not
Thus we i1ave the following:
~~~g~~~=~~=~.
•
Let
t ~ 2 be an integer.
q be a prime-power and let
either (i) there exists a non-Desarguesian translation plane of order
•
(ii)
q
=2
Proof.
•
and t
d = 1, s = t, Q = q.
we may apply Theorem 13. 2 with Q = q
d
If
d
.,g
q = 2
> 2.
= 2
is a t-spread over GF(q), then
t
q.
the translation plane defined by .-.& has order
If
and
q
> 2 we may apply
t = ds wit h
which is not Moufang.
e
cat!ve group of
and with
e.
...6'
over
This proves Theorem 13.2.
We may add the remark that if
holds with Q - 1 = 1
d? 2, s ? 2,
In either case, by the remarks
in the paragraph just preceding Theorem 13.2, there exists at-spread
GF( q)
or
is a prime•
First we recall, from (1), that it
Theorem 13.1 with
t
q
Then
q = 2
and
t
is a prime, then (13.3) still
?Ji = %1 corresponding to the whole mu1tip1i-
But then we have no means of constructing a non-Desarguesian
-ef, even though we have a perfectly good
spread from the given Pappus spread
swltching set in ~ .
•
~~=~~;,~g~~. As an illustration of the power of Theorem 13.1, we shall
14.
discuss planes of order
We can take
d
= s = 2,
4 in terms of 4-spreads over GF(3).
3
Q = 9,
Consider the matrices (over
X
=
f~
\0
,1
1
0
0
0
1
0
0
0
1
1
0
0
~(s) = 1, or
Here
d = 1, s = 4, Q = 3,
q=3, t=4.
~(s) = 2.
GF'(3»
,
T
=
:'1
0
0
0
fo
0
0
1
'0
0
1
1
\0
1
2
1
These satisfy the relations
"
_
x4 = I
where
I
+ X
,
T- l X T =
x3 ,
is the 4-rowed identity matrix.
multiplicative order
4
3 _1
= 80.
,
The matrix X is irreducible a.nd has
Thus (in matrix form) the collection
E con-
75
sisting of the zero matrix
0
and the 80 powers of
represents a Pappus spread ~ over
order
•
•
10 generated by
T ~
2
-13.
GF(3).
represents a conjugate switching set.
2
and each of T ~
T3~
,
'<
a field
c:e
Then
GF(3
4) which
be the multiplicative group of
Then % represents a switching set of
group of order 40 generated by X •
Again, let
---8,
and
d5 be the multiplicative
repre sents a switching set of
4,
represents a conjugate switching set.
Let us note the cosets, formed from
and
Let
X is
~, which are contained in
,;t,
T £.
T3,;e
,
0<i<3
= =
o<
=
<
3
=
i
,
For a given
i,
T2 :x, xi; and T
'DLX
S)-(.
i
Xi
is a (representation of a) switching set with conjugate
is a switching set with conjugate switching set
T3~~ Xi.
Now let us note that the following collections represent spreads:
•
e;'
Here
t::
o ,
£,
;eX
o ,
et:,
T.;£X
is Desarguesian (or Pappus) but
by expressing
.:c
and
~,
,;r: X in terms of cosets
is not.
~ x2i
We may analyse
e
and ~ Jfi+l.
further
If o/'J'l,
is any of these 8 cosets except % itself, we may derive a new representing
•
collection from
e
by replacing
2
t:?Y11J by T '?;'l,..; and we may make up to 7 such re-
i
placements simultaneously. Similarly,
't::' contains cOlets X
and
i l
T X
+ • If ~ is any of these 8 cosets except % itself, we may derive a
x2
x2
~, by replacing ~ by
new representing collection from
•
we
can make up to
7
such replacements simultaneously.
7
2.2
collections representing spreads.
=
8
2
=
256
2
T
"12 ; and,
In all we get
again,
Note that we have been discussing representations (9.1) 1dth t
•
~
The
27 collections obtained from ~ can be distinguished from the
tions obtained from
#t
v
on the following basis:
the former have representations (9.1) with t
to the latter collections do not.
= 2,
= 4,
q
= 3.
27 collec-
The planes corresponding to
q
= 9;
the planes corresponding
The planes corle spending to all of the 28
collections do have something in common, l:owever, as the :reader may verify by conj
i
sidering the role of a s~dtching set corresponding, for example, to T . ~ X •
Namely, all of the planes have systems of (tranSlation) subplanes of order 9 to
which may be applied the process of Ostrom for planes of order 92• (See the discus sion in [1].)
If
~
C1
8
is anyone of our 2
C 1 determine s an equivalence
from ~l by tl~ processes of
collections,
class consisting of all collections obtainable
inversion, affine transformation and conjugation.
•
It would be of interest to have
a system of representatives for these equivalence classes - and perhaps also
system of representatives for the subclasses under conjugation.
Then one
a good start on the problem of determining all translation planes of order
The discussion of translation planes of order p4,
analogous to the above for
.
•
•
•
p
a
v~uld
have
4
3.
p prime, is completely
p odd, and corresponds to the case
t
= 2, q = 4 for
= 2•
Similar studies may be w..ade of translation planes of order pt
a fixed positive integer and p
is a variable prime•
where
t
is
- 77
Added July dO, 1964.
April 15, 1964.
The foregoing was turned over 'to 'the t;ypis't abou't
In le'tters to Bruck dated June 14 and June 19, 1964, respectively,
G\mter Pickert and Wilbur Jonsson pointed out the close connection between our
;
paper [1] and an earlier paper of Andre [6).
(see alao Pickert [7], pp. 199-220,
eSPecially pp. 199-200.)
We may interpret AndJ' a construction briefly as follows:
Let t be a positive
integer, V be a 2t-dimensional vector space over a skew-field F.
Let K be a
collection of t-dimensional vector subspaces of V over F such that each nonzero
vector of V lies in exactly one member of K.
The points of n are the vectors in V.
is in K and v is in V.
V.
Define a system n = n(K) as follows:
The lines of n are the coaets X + v where X
The incidence relation of n is 'the containing relation of
Then n is a transla.tion plane.
Moreover, a translation plane n can be re-
presented in form n(K) precisely when n can be represented by Bose 1 s construction.
(see section 9 of the present paper).
I
It should be observed that the Andre and Bose constructions are qUite different.
The former uses a vector space of even dimension 2t.
jective space
2t.
--
The latter uses a pro-
of even dimension 2t (and hence a velttor space of even dimension
The latter uses a projective space of even dimension 2t.
The latter uses a
projec'tive space of even dimension 2t (and hence a vector space of odd dimension
•
2t + 1.)
On the other hand, both constructions depend upon a spread.
/.
/
As Andre
/
himself points out, the K of Andre's construction (called by Andre a congruence)
is, in our terminology, the vector space form of a spread of (2t - l)-dimensional
projective space.
Thus the statement in [1]-- that spreads are used for the first
time in [1] for the geometric construction of planes-- is debatable.
•
The theory of spreads is, of course, equally important for both constructions •
In partiCUlar, Theorem 13.1 of the present paper is impl1citly given by Andre.. .
( (6] , p. 184.)
We should like to apologize here for a typographical blunder in [1]:
consistently mSSteIled the name OSTROM.
We have
78
,
,
BIBLIOGRAPHY
[1]
R. H. Bruck and R. C. Bose.
projective spaces.
1
•
[2]
Irving Kaplansky.
The construction of translation planes from
Journal of Algebra,
1
(1964), 85-102.
Rings with a polynomial identity.
Bull. Amer. Math. Soc.
54 (1948), 575-580.
[3] Marshall Hall, Jr.
229-277
Projective planes.
Trans. Amer. Math. Soc.
2!!:.
(1943),
and §2. (1949), 473-474.
[4]
Marshall Hall, Jr.
[5)
R. H. Bruck.
The Theory of Groups.
MacMillan, New York, 1959.
Recent advances in the foundations of Euclidean plane geometry.
Amer. Math. Monthly ~, No.7, II (Slaught Memorial Paper No.4), (1955),
2-17.
[6]
Johannes Andre.
Gruppe.
(7)
•
•
,
•
fiber nicht-Desarguessche Ebenen mit transitiver Translations-
Math. Zeit. 60 (1954), 156-186.
"
Gunter
Pickert. Projective Ebenen.
Heidelberg, 1955•
Springer-Verlag,
Berlin~ttingen-