0 7 8 9
6 1 7 8
5 0 2 7
1+ 6 1 3
0
7
8
9
1
3
5
2
1+
6
6
1
7
8
9
2
1+
3
5
0
1
9
8
7
3 5 2 1+
2 1+ 3 5
9 3 1+ 6
8 9 5 0
8 6 1
5 1+ 9 8 7 1 2 3
7 0 2
0 6 5 9 8 2 3 1+
6 1 3
2 1 0 6 9 3 1+ 5
0 7 8
7 3 2 1 0 1+ 5 6
1 9 7
8 7 1+ 3 2 5 6 0
2 8 9
9 8 7 5 1+ 6 0 1
3 9 8 7 6 0 1 2
1+ 5 6 0 1 7 8 9
6 0 1 2 3 8 9 7
1 2 3 1+ 5 9 7 8
6
0
1
2
3
1+
5
9
8
7
COM BIN A TOR 1 A L
MAT HEM A TIC S
YEA R
Februal'y 1969 - June 1970
LINEAR REPRESENTATION OF A CLASS OF PROJECTIVE PLANES IN
FOUR DIMENSIONAL PROJECTIVE SPACE *
by
R.C. Bose
University of North Carolina
at Chapel Hill
and
A. Barlotti t
University of Perugia,
Italy
Institute of Statistics Mimeo Series No. 600.20
Department of Statistics
University of North Carolina at Chapel Hill, 27514
FEBRUARY, 1970
*
This research was sponsored by the National Science Foundation under
Grant No. GP-8624, and the U.S. Air Force Office of Scientific Research under
Grant No. AFOSR-68-1406,
This research was conducted while the author was visiting professor at
the University of North Carolina at Chapel Hill, His research was also
partially supported by C.NoR,
t
LINEAR REPRESENTATION OF A CLASS OF PROJECTIVE PLANES IN
FOUR DIMENSIONAL PROJECTIVE SPACE *
by
t
R.Co Bose
University of North Carolina
at Chapel Hill
A. Barlotti
University of Perugia,
Italy
ABSTRACT
The theory of linear representations of projective planes was developed
by Bruck and one of the authors (Bose) in two earlier papers [J. Algebra
1 (1964), 85-102 and 4 (1966), 117-172] can be further extended by generalizing the concept of incidence adopted there.
A linear representation is
obtained for a class of non-Desarguesian projective planes illustrating
this concept of generalized incidence.
It is shown that in the finite
case, the planes represented by the new construction are derived planes in
the sense defined by Ostrom [Trans. Amer. Math. Soc. 111 (1964), 1-18] and
Albert [Boletin Soc. Mat. Mex., 11 (1966), 1-13] of the duals of translation
planes which can be represented in 4-space by the Bose-Bruck construction.
An analogous interpretation is possible for the infinite case.
* This research was sponsored by the National Science Foundation under
Grant No. GP-8624, and the U.S. Air Force Office of Scientific Research
under Grant No. AFOSR-68-1406.
t
This research was conducted while the author was visiting professor at
the University of North Carolina at Chapel Hill. His research was also
partially supported by CoN,R.
2
I.
INTRODUCTION
Bruck and one of the authors (Bose) in two earlier papers [2], [3],
developed a theory of linear representation of projective planes, in which
a projective plane
of
8
8
in at-dimensional (Desarguesian) projective space,
2
each point or line of
jective subspace of
in
8
t
(Desarguesian or not) is an isomorphic representation
2
c
8
in which
becomes a non-empty finite dimensional pro-
2
8 ,
t
8 ,
t
and incidence is given by the containing relation
In particular an explicit linear representation was obtained for
all translation planes cooridnatized by a right Veblen-Wedderburn system
which is finite dimensional over its left operator skew-field.
eludes the class of all finite translation planes.)
further step in the same direction.
The present paper is a
It now appears that the earlier def-
inition of incidence based on the containing relation in
restrictive.
8
In general the incidence between a subspace
a line, and a subspace
S
(This in-
t
a
is too
representing
representing a point can be defined by spec-
ifying the dimension of the projective space
a n B
A linear representation is developed for a class of projective planes
6,
illustrating this concept of generalized incidence.
of
~,
are represented by certain subspaces of a projective space
~
four dimensions.
~
Points and lines
is a fixed 3-space
8 ,
4
S) and
space, intersecting
L
Points of
~
~
is a fixed line of
in a plane containing
is contained in
S.
~
is a fixed 3-
~.
and passing through lines of
(2) those represented by planes of
~;
~
are of three types, (1) those represented by planes of
not contained in
passing through
of
4
S is a spread of lines in
(i.e., a set of lines, such that each point of
exactly one line of
8
S,
8 , not contained in
4
other than
L
or
~
(3) those represented by the points of the line
~
and
~.
,.
3
~
Lines of
of
S4
are of three types, (1) those which are represented by points
not contained in
planes contained in
by the line
in
S4'
E
or
Q; (2) those which are represented by
~;
Q, and not containing
(3) a single line represented
Incidence is defined by the containing contained relation
~,
except that a point of type (1) or type (2) is defined to be in-
cident with a line of type (2), when the planes representing them intersect
in a line.
dence
~
It is shown in Section 3, that with this definition of inciis a projective plane when
infinite and
S
S4
is a finite space or when
has the property that each plane of
L
contains not more
In Section 4, the relation between translation planes and
A translation plane
T,
is
S.
than (and therefore exactly) one line of
studied.
S4
~-planes
is
which can be coordinatized by a right
Veblen-Wedderburn system which is two dimensional over its left operator
skew-field may be said to be of degree two.
projective space
S4
in the manner described in Section 2 (being a special
case of the construction given in [2]).
represented by a particular 3-space
T.
Then
the
~-planes
L
It can be represented in a
represents a point of
The line at infinite in
L of
aT,
S4'
L),
aT
be the dual of
are the derived planes (in the sense defined by Albert [1] and
collinear with
L
cT.
Any two points of
can be embedded in a derivation set (which includes
An analogous construction is possible in the infinite case
be noted that
for any line
is
It is shown that in the finite case
Ostrom [10], [11], [12]) dual translation planes
aT
Let
T
E
~
is a point in
through
6,
aT,
such that
aT
is
(E,~)
It should
transitive
and is unique except for the cases when
is a Moufang plane or is Desarguesian, when
E
aT
is arbitrary.
In Section 5, we show that the process of dua1ization and derivation
by which the plane
~
is obtained from
T
can be continued if the line
4
-6
of the spread
S,
belongs to a regulus all of whose lines are in
S.
Thi.s gives the geometrical representation of the analogous sequences of
planes obtained by Fryxell [7] and Johnson [8].
lation plane
T
= TIl' we obtain a cyclic sequence
of projective planes such that
and
TI
+
2i l
Starting from the trans-
is the derived of
TI
TI
=
i
Zi
'
TI i +8 ,
{TIi1.
(i
= 1, 2, 3",)
is the dual of
TI .
21.-1
5
2. TRANSLATION PLANES
1.
Let
F
be a skew field (that is, an associative division ring which
mayor may not be commutative).
space based on
F.
Let
The skew field
8
F
be a four dimensional projective
4
may be finite or infinite.
be specially interested in the first case, when
GF(q)
of order
q,
where
Z,
of lines contained in
exactly one line of
S
a line
~
S.
must be a Galois field
is a prime power.
Z be a fixed 3-space of
Let
line of
q
F
8 ,
4
A spread
S is defined as a set
Z is contained in
such that every point of
A plane
IT
in
E
cannot contain more than one
(otherwise they would intersect in a point).
belonging to
a unique point
8'
S,
then every other line
, and through any point
passes a unique line of
S.
lines, and any plane of
E
We shall
of
8'
In the finite case
S
If
contains
IT
S meets
~'
of
IT ,
not in
S,
there
contains exactly
contains exactly one line of
S.
in
IT
2
q +1
For further
properties of spreads see references [2], [3], [4], and [13].
2.
The following is a special case of a construction given in [2) for
obtaining translation planes.
be denoted by
T
The translation plane to be constructed will
and its points and lines will be called T-points and
T-lines, to avoid confusing them with elements of
T-points are of two types.
the points of
8
4
of
S.
T-1ines are of two types.
8 ,
4
which represent them.
E.
In the finite case there are
q
4
T-points of type (2) are represented by the lines
In the finite case there are
planes of
4
T-points of type (1) are represented by
not contained in
T-points of type (1).
8
q2+l
T-points of type (2).
T-lines of type (1) are represented by
passing through the lines of
S and not contained in
L.
6
In the finite case through each line of
contained in
L,
so that there are
S there pass q 2 planes not
q4+q2
T-1ines of type (1).
There is
only one line of type (2), which is represented by the fixed 3-space
To complete our description of the plane
between T-points and T-1ines.
only if the element of
54
T,
we must define incidence
A T-1ine and a T-point are incident if and
representing the T-1ine contains the element of
4
representing the T-point. It was shown in [2], that the T-points and
5
T-lines form a translation plane, which is Desarguesian if and only if
is a regular spread.
The spread
S
-6 , -6 , -6
2
1
3
generators (ruling lines) of the regulus to which
elements of
2
q •
S.
S
is said to be regular if for any
hyperbolic quadric containing three lines
is
Lo
of
S
-6 1 ' -6 2 , -6 3
all the
belong are
In the finite case the order of the translation plane
T
7
3.
1.
li-PLANES
We shall now obtain a new projective plane
described in Section 2.
The four space
S of lines in L,
the spread
L
related to the T-plane
the fixed 3-space
remain as in Section 2.
case we shall additionally assume that
plane of
S4'
li,
li
and
In the infinite
S has the property, that every
S.
contains at least (and therefore exactly) one line of
noticed earlier this is always true in the finite case.
lines of
L,
will be called li-points and li-1ines.
struct an affine plane
ali
As
The points and
We shall first con-
and later complete it to a projective plane
The relationship between the planes
T
and
li
li.
will be considered in
Section 4.
~
Let
of
S4
~
which passes through
containing
S4'
are not contained in
~,
ali-lines are two types.
S4
not contained in
5
(q -1) / (q-l)
points,
S
q
L
which is contained both in
2
in a plane
~,
other than
In the finite case there are
through each of which pass
L
IT
They are represented
which pass through a line of
L.
be a fixed 3-space
and therefore intersects
q
2
lines of
and
S
L.
planes not contained in
4
Hence the number of ali-points is
of
~
Let
ali-points are all of the same type.
~.
by the planes of
other than
S.
be a fixed line of the spread
q •
ali-lines of type (1) are represented by points
or
L
and
L
In the finite case
~.
and
is the number of type (1) ali-lines.
~,
3
(q -1) / (q-l)
has
~
the number of points contained neither in
resented by planes contained in
4
(q -1) I (q-l)
each have
~
~
has
S4
points, and
points.
L is
nor in
IT
Hence
4 3
q -q .
This
Again ali-lines of type (2) are repnot passing through
case the number of planes contained in
~
is
4
.
~.
(q -l)!(q-l)
In the finite
of which
8
(q2_ 1 )/(q-l)
is
(R)
~he
~o
pass through
q 3+q 2
a~-lines
The total number of
be the set of points of
set of planes of
SI
by planes of
S4
(R),
not contained in
and
of type (2)
4 2
q +q
in the finite case is
~o
not containing
represented by points of
a~-lines
Hence the number of
or
a~-lines
Then
a~-lines
L:
SI,
and
of
~ype
Let
0
(P)
be
tl) are
of type (2) are represented
(P).
To complete the description of the
a~.
between points and lines of
a~-plane
we have to define incidence
An a~-line of type (1) is said to be in-
cident with an aL1-point, if the point representing the former is contained
in the plane representing the later.
a point
Q of
(R),
a~-line
case every
a~-points
then the
by the planes joining
Thus if an
Q to the lines of S,
intersects
IT
in a line
section of
l
and
a~-line
~.
lather than
If
0
l
in a point
must contain the unique line
is the plane through
~,
be a point of
lying in
~
S',
since
~'
~,
s' .
of
containing
~I
and
IT
S' .
Let
6
then
L
(P),
~
then
be the point of inter-
intersects
Hence
S
~
~n
a line
SI
passes through
which passes through
~'
,
Thus
S'
cannot
~.
Hence to each line
~)
(i.e., not meeting
a~-point
m
there corre-
incident with the a6-1ine
It is obtained by finding the intersection
ill
6
S'.
is disjoint from
L
m,
and
Note that
and taking the plane containing
passes through
if the planes repre-
be a plane of
~.
a~-line
m,
sponds a unique plane representing an
~.
a~-point
An
a~-points.
and
and not passing through
represented by
2
q
In the finite
is a plane representing an a6-point incident
represented by
which intersects
~
Let
~.
other than
of type (1), is incident with
senting them intersect in a line.
is represented by
incident with it are represented
of type (2) is said to be incident with an
with the
a~-line
and the line
In the finite case there are
2
q +q+l
~'
S'
of
of
S
lines in
ill
which
iJ ,
9
of which
q+l
with exactly
pass through
q
2
L.
Hence any
of type (2) is incident
a~-points.
a~-lines
Let us consider the set of
~.
incident with a given
° passing through a
represented by a plane
contained in
a~-line
a~-lines
line
~' ~ ~
°
Also
~
intersects
passing through
m,
types.
in a line
0.
not meeting
~'
passing through
or
m.
(R)
m belongs to
q+l
q -q
(P),
~.
a~-point
The planes of
(P)
a~-
is incident with lines of both
contained in
0,
such points.
Also every plane
is incident with
of type (2).
0,
lying in
are the points
since no such plane can contain
a~-point
Hence in the finite case each
of type (1) and
2
There are
(R)
and not
of type (2) incident with the
a~-point
Thus each
In the finite case points of
not lying in
q2_q
m,
a~-lines
represent
point represented by
S,
of
of the first type incident with the
under consideration are represented by the points of
a~-point,
~.
a~-lines,
q2+l
We now proceed to show that
a~
is an affine plane.
2.
Lemma (1).
Two distinct a~-points are incident with a unique a~-line.
We have to consider two separate cases:
Case I.
Let the planes
02,
~',
are points of
of (R) common to both
incident with both
planes of
(p).
01
01
and
m
2
II
in the point
through
S'.
representing the two
of
S.
Since the points common to
02.
02.
S' •
S
~,
Hence there is no
a~-lines
Now by construction
Hence every other line of
meet
02
which is contained in
and
meets
Now
a~-points,
and
~'~~,
pass through the same line
and
01
01
IT
IT
there is no point
a~-line
in a unique point.
There is a unique plane
of type (1),
of type (2) are represented by
contains the line
and
°
°
2
~
meet
=
rl
(m , m )
2
l
~
of
Let
in lines
of
(P),
S.
~'
and
which
10
and
contains
This plane intersects
a~-line
represents the unique
sented by
Case 110
which is incident with the
°
a line
and
°
Let the planes
and
1
Then
u.
-6
62
would meet
u
This is contradiction since
-6
and
-6
Q cannot belong to
(-6
2
1
;' -6, -6
2
'f -6)
U
common to
are disjoint.
2
0
-6
and
1
S.
-6
2
,
But two planes in
Let this point be
Q
st.
But it may or may not belong to
L
of
Suppose if possible they meet in
in a point
L
1
-6
4-space must have at least one point in common
Clearly
repre-
representing two distinct a6and
1
cannot meet in a line.
02
a~-points
2'
points, pass through different lines
Now
in lines and
°
and
0,
and
Hence
we have to consider two subcases:
(a)
set
and represents an
(R),
with the
a~-points
(b)
IT
Sl
represented by
and
S2
S152
lying in
(P),
in
and
u
IT
=
(m ,m )
l 2
and
S1.
m
2
and represents an
Hence
=
-6.
-6
1 and
Hence
-6
o}
-6.
inter-
02
de-
and
and intersecting
Hence
S meet
of
and
The lines
S1
2
L in the line
u is an element of the
of type (2).
It intersects
and
0,
represents the unique a6-line incident with
jJ
the a6-points represented by
02.
The lines
lying in
a~-line
Then it is an element of the
which is the unique a6-1ine incident
01
and distinct from
and
S1.
not contained in
in lines
termine a plane
a~-line,
Q belongs to
Suppose
in points
sect
set
Q does not belong to
Suppose
01
and
02n
This completes the proof of
Lemma (1).
3.
The a6-lines can be divided into parallel classes.
are of two kinds.
not lying in either
Let
L
(P*)
or
Parallel classes
be the set of planes passing through
S1.
In the finite case
(P*)
-6
consists of
and
11
2
planes, since through
q -q
exactly
of which lie in each of
L
and
including
S"i,
2
there pass
-6
q +q+l
q+l
planes,
which lies in both.
II
Two aL'.-lines of type (1) , represented by the points
and
Ql
Q2
of
(R)
are defined to belong to a parallel class of the first kind, i f both
Q
and
are contained in a plane
Z
A of the set
of type (1) represented by
Q
and
l
This parallel
(p*).
A.
class will be said to correspond to the plane
Q1
It is clear that
a~-lines
Q are parallel (belong to the same
Z
Q1Q Z meets
parallel class) if and only if the line
L,
in a point
-6
in which case they belong to the parallel class corresponding to the plane
A which contains the line
LQ1Q .
Z
and the line
-6
Every
a~-line
(1), belongs to a unique parallel class of the first kind.
point of
(R),
which the
a~-line
taining
Q and
A of
Z
q -q
(P*),
con-
of type (1) can thus be partitioned into
In the finite case the
partitioned into
Lemma (2).
a~-lines
-6.
Q is a
If
then the unique parallel class to
belongs is represented by the plane
parallel classes.
one or zero
a~-line,
representing an
of type
4 3
q -q
of type (1) , are
a~-lines
parallel classes each containing
q
Z
members.
Two a~-lines of type (1) are simultaneously incident with
a~-points
according as they are non-parallel (belong to dif-
ferent parallel classes) or are parallel.
Consider two
Q of
Z
(R).
If
a~-lines
8
is a plane representing an
incident with both the
a~-lines,
in
Q1Q
Z
and the line
-6'
of
-6,
then the
a~-lines
the
a~-point
represented by
Ql
of type (1), represented by the points
S'.
then
a~-point
8
8
Q
l
Q1Q Z.
must be the plane containing
S which passes through
represented by
simultaneously
8 passes through the line
Then
and
and
S'.
If
S'
is not on
QZ are non-parallel and
is uniquely determined.
However if
S'
12
is on
i.e., the
~,
a~-lines
S
then the only line of
~
do not represent any
represented by
through
SI
a~-point.
taneously incident with any
is
Q
l
~.
and
Q
2
are parallel,
By definition planes through
a~-lines
Hence the two
are not simul-
On the other hand they belong to
a~-point.
the unique parallel class of the first kind corresponding to the plane
of
containing
(P*)
Let
QlQ
Z
~.
and
denote the set of points on
(0)
consists of exactly
by the planes
q+l
and
WI
points.
W2
of
Two
~,
of
L.
(0).
~.
a~-lines
In the finite case
WI
and
~,
are defined to belong
W2
meet
in the same
~
This parallel class will be said to correspond to the point
Every
a~-line
the second kind.
If
by the point
L
W is a plane of (P)
(0),
of
in which
representing an
a~-line
W intersects
of type (2), are partitioned into
each containing
Lemma (3).
one or zero
q
2
a~-line,
then
corresponds is represented
(Z) can thus be partitioned into parallel classes.
a~-lines
L
of type (2) belongs to a unique parallel class of
the unique parallel class to which the
q 3+q 2
(0)
of type (2), represented
not containing
to a parallel class of the second kind if
point
A
~.
a~-lines
of type
In the finite case the
q+l
parallel classes,
members.
Two a~-lines of type (2) are simultaneously incident with
a~-points
according as they are non-parallel (belong to dif-
ferent parallel classes) or are parallel.
Consider two
WI
and
W2
the line of intersection of
WI
and
W2.
lie in
(P).
of type (2) represented by the planes
By definition
W2
of
a~-lines
IT.
Then
m meets
IT
in a point
cannot contain
~.
Let
Suppose first that
SI.
A plane
0
~l
and
m be
m does not
representing
13
an o,6-point incident with both the all-lines represented by
m and the line
must contain
is not on
W2
which passes through
S I,
If
is uniquely determined,
0
i.e., the alllines represented by
S
and
i.e. , the al':.-lines represented by
-6 ,
parallel, then
line of
S
of
-6 '
u1
S'
through
is
represent any al':.-point.
-6.
Lri
and
]..11
and
are non-
iJ2
SI
I f however
is on
are parallel, Lhen the only
02
By definition planes through
On the other hand they belong to the unique
the line of intersection of
m is different from
Then
do not
-6
SI
parallel class of the second kind corresponding to the point
m,
-6 ,
Hence the two all-lines are not simultaneously in-
cident with any all-point.
Next suppose that
SI
and
]...1
and meets i t in a point
-6 ,
the two all-lines represented by
Lrl
and
of
(0) •
lies in
u ")
~
n.
In this case
L.
belong to the unique parallel
Lr2'
class of the second kind corresponding to the point
L
(0).
of
We shall
show that there is no all-point simultaneously incident with these all-lines.
If possible suppose such an all-point exists and is represented by the plane
o.
Then the line of intersection of
0
Lr2
and therefore coincides with
Hence
Therefore
m belongs to
only line of
S,
S,
m,
~
and
0
must lie in both
intersects
in
L
which contradicts the fact that
which lies in
and
WI
m.
is the
-6
IT.
Finally two all-lines of different types are defined to be nonparallel.
The appropriateness of this is apparenL from the following
Lemma.
Lemma (4).
Two a6-lines one of which is of type (I), and the other of
type (2), are simultaneously incident with a unique a6-point,
Consider an all-line of type (1) represented by a point
and an all-line of type (2) represented by a plane
]..I
of
Q of
(P).
Let
(R),
¢
be
14
the 3-space containing both
Q and
is distinct from
Hence
or
S6
Lo
only 3-space containing
n*
not ,::.ontaining
is distinct from
-6
Then
<
If
-6.
and
~
6
n~c
in a line.
u
S.
pass through a line of
u.
does not contain
Hence
1>
Since
'"
and
Hence
Q,
then
for
S6
L
1>
is the
which
-6,
incident:
~8-po1nt:
1>
2:,
in a plane
Also
6
.t is the only line of S in
Q
must
¢,
6 is
.t and Q
Lemmas (2), (3) and (4) taken together show that
a~-point
or
must contain
6
lies in
cS
S6
.t of
contains a unique line
the uniquely determined plane containing
cident with no
~,
intersects
1S a plane representing an
with the 0.o-1ines represented by
and must intersect
1>
Q is not: in
Since
Cl,
LWO
a6-lines are in-
if they are parallel and with a un1que a8-point if
they are non-parallel,
4.
Lemma (5).
Given an a6-line and an a8-point not incident: with it,
there exists a unique a6-line parallel to the given a6-line, and incident
with the given
a~-point.
Let the given a6-point be represented by the plane
a line
S
of
-6'
(-6~-6'),
6
By definition
6,
pass1ng through
is not conralned in
L
We
have two cases to consider.
Let the given a6-line be of type (1), represented by the point
Case -I.
Q of
-6
(R) .
and
Q.
Then
Q is not contained in
Q'
of
(P).
A
be the plane joining
A,
Clearly
,\
intersects
6
in a unique
which represents the required a6-1ine,
Case II.
u
Let
Lines of the parallel class to which the given aLi-line belongs
are represented by the points of
point
6,
Let the given a6-1ine be of type (2), represented by a plane
Then
6
does not intersect
~
in a line.
Let
L
be the
15
w and
point of intersection of
6
and
~
~,
does not lie in
Then the line of intersection
~.
and does not pass through
a6-line is represented by the plane containing
Lemma (6).
Land
L.
m of
The required
m.
There exist three a~-points not incident with the same
a~-line.
Let
be a point of
and let
(R) ,
and
~3
respectively.
(R)
on
03,
represents an
Then the
same
a~-points
~-line.
Ql'
in some order or other.
~
(i
= 1,2,3).
joining
°3
Q
3
to
~2
Q
2
and
~l
different from
Ql'
are not incident with the
represented by
where
Q
l
Choose a point
a~-points.
°1, °2, °3
a~-points
Qi'
f
in a point
°2
represented by
dent with the unique line
1
Then the plane
and meets
In fact the
~.
be the planes joining
°2, 03
They represent
other than
a~-point,
S,
be three lines of
~ l' ~2' ~ 3
i, j , k
°.J
and
ok
are the indices
This completes the proof that
are inci1, 2, 3
is an affine
a~
plane.
5.
The completion of the affine plane
is now straight forward.
a~-lines
~-points
We now define new
to the projective plane
may be called
of types (1) and (2) may be called
respectively.
Thus
a~-points
a~
~-points
~-points
~-lines
we have to define a new
~-line
of type (1).
of types (1) and (2)
corresponding to parallel classes.
of type (2) are represented by planes of
of type (3) are represented by points of
(0),
(P*),
types (2) and (3), which may be represented by
and
i.e., points of
of type (3) incident with all
plane may be described as follows.
~
~.
~-points
~.
~-points
Finally
of
Hence the completed
Points and lines are of three types:
~
16
Points of type (1) are represented by planes, passing through the
lines of
S
other than
~,
and not contained in
E.
Points of type (2) are represented by planes passing through
not contained in
~
or
and
60
~.
Points of type (3) are represented by points of
Lines of type (1) are represented by points not contained in
Lines of type (2) are represented by planes contained in
containing
~,
~,
E
or
and not
~.
~.
There is a single line of type (3) represented by
A 6-line of type (2) is incident with a 6-point of type (1) or (2) if
and only if the planes representing them intersect in a line.
In every
other case a 6-line is incident with a 6-point if and only if the element
of
8
8
representing the 6-point.
4
~
4
representing the 6-1ine is contained in or contains the element of
Thus the 6-line of type (3) represented by
is incident with every 6-point of type (2) or (3) and non-incident with
any 6-point of type (1).
A,
A 6-point of type (2) represented by the plane
is contained in
of type (2).
~.
A 6-point of type (2) is not incident with any 6-1ine
A 6-point of type (3) represented by a point
~,
with a 6-1ine of type (2) represented by the plane
through
(1).
Q representing it
is incident with a 6-line of type (1) if the point
L.
if
L
is incident
~
passes
A 6-point of type (3) is not incident with any 6-1ine of type
The incidence between 6-points of type (1), and 6-1ines of type (1)
or (2) is carried over from the affine plane
In the finite case the order of
types (1), (2) and (3) respectively is
6
is
q
4
a6.
q
2
.
, q 2-q
lines of types (1), (2), (3) respectively is
4
q -q
The number of points of
and
3
,
q+1.
3
q +q
2
The number of
~d
1,
~.
17
4.
1.
DERIVATION OF ~-PLANES FROM DUAL TRANSLATION PLANES
In this section we shall consider the relation between translation
~-planes
planes described in Section 2, and the
Let us start with the translation plane
obtain a dual translation plane
aT
obtained in Section 3.
T.
We can dualize it and
by taking the T-lines for aT-points,
and the T-points for aT-lines.
Thus we have a 4-space
S
is a spread of lines in
S4'
E.
together with a fixed 3-space
Then
aT
S4
not contained in
S.
lines of type (2) are represented by lines of
aT
E,
aT
lines
S.
and passing through the lines of
E.
aT
points are of two
aT-points of type (1) are represented by planes of
tained in
S4'
lines are of two types.
of type (1) are represented by points of
types.
E of
S4
not con-
There is a single
aT-point of type (2), represented by the 3-space
E.
A aT-line and a oT-
point are incident if and only if the element of
S4
representing the oT-
line is contained in the element of
aT
is a dual translation plane.
a line at infinity.
a aT-line.
E,
S4
Let
~
representing the aT-point.
We can make it affine, by choosing
be a line of the spread
The aT-points incident with
and by the planes of
S4'
aT.
Then
~
represents
are represented by the 3-space
passing through
We will choose the aT line represented by
in
~
S.
~
~
and not contained in
E.
to be the line at infinity
The aT-points not incident with the line at infinity, may be
called aoT-points, and the aT-lines other than the line at infinity may be
called aoT-lines.
over from
aoT
aT.
The incidence between aoT-points and aoT-lines is carried
Thus the elements of the affine dual translation plane
are as follows:
18
aoT-points are all of one type.
S4'
S
passing through the lines of the spread
4
finite case their number is
S,
They are represented by the planes of
other than
aoT-lines are of two types.
of
8
of
S
~.
other than
(1) and
q
2
q
2
planes of
8 ,
4
acT
q
4
acT-lines of type
can be divided into parallel pen-
All aoT-lines passing through a given oT-point
oT
belong to the same parallel pencil.
oT-point at infinity with which all
aoT
2
case the number of parallel pencils is
P2'
identify
be two planes of
Then
tained in
and
oT.
Section 3.
pass through
and
Let
~.
and
Let
and let
IT
q +1,
In the finite
each parallel pencil con-
aoT-lines.
PI, P 2
infinity in
The
lines of a parallel pencil are
incident will be called the vertex of the parallel pencil.
Let
~.
aoT-lines of type (2).
on the line at infinity in
2
lines of
not contained in
In the finite case there are
cils in the usual manner.
q
2
aoT lines of type (2) are represented by lines
The lines of the affine plane
taining
q
In the
aoT-lines of type (1) are represented by points
~.
not contained in
4
since through each of the
q ,
there pass
~,
~.
other than
IT
Q
(D)
~
::;'2
S4'
~
passing through
and not con-
represent two oT-points of the line at
be the 3-space of
S4
~
be the plane of intersection of
and
~.
denote the set of planes of
In the finite case
L,
We shall
with the spaces denoted by the same symbols as in
(D)
~
other than
consists of
q
will be called the derivation set.
case the derivation set consists of
q+l
IT,
which
planes.
The set of oT-points at infinity represented by the planes of
and the 3-space
~
containing the planes
(D),
In the finite
oT-points at infinity.
Each oT-point of the derivation set is the vertex of a parallel pencil
of the affine plane
aoT.
We shall now construct a new affine plane from
19
aoT
in the following manner:
The points of the new affine plane will be
the same as the aoT-points.
The aoT-1ines belonging to the parallel pen-
ci1s whose vertices are oT-points of the derivation set will be deleted.
In what follows we shall refer to these as the deleted lines, and the pencils formed by them as the deleted pencils.
The deleted aoT-lines will be
replaced by new lines, which are certain (affine) Baer subp1anes of
Before proceeding to consider these, we shall make a
between aoT-points and the lines of
2.
not intersecting
~
Any aoT-point is represented by a plane
~'
of
~,
otherwise
S
~),
(other than
unique line
0
m,
~.
would be contained in
If
Sf.
Through
S'
aoT-point corresponding to
I
3.
of
and
Let
~,
~.
passing through a line
~.
Hence
Then
0
0
o.
~
in a
m
This correspondence is
~,
then
passes a unique line
~'
of
m,
cannot inter-
intersects
not intersecting
m is any line of
in a point
~
n
correspondence
We shall say that this line
~.
corresponds to the aoT-point represented by
(1,1).
0
and not contained in
not intersecting
(1,1)
aoT.
m meets
S.
is represented by the plane
IT
Then the
0
containing
m.
)1
be a plane of the set
not passing through
~
.
(P)
of Section 3, i. e. ,
We shall show that
representing an affine subplane of
points corresponding to the lines of
aoT.
)1
)1
is a plane
)1
can be regarded as
Consider the set
not intersecting
(B )
of
)1
~
.
aoT-
This we de-
fine to be the set of aoT-points belonging to the subplane represented by
)1.
To each line
m in
sented by a plane
and intersecting
there is a unique aoT-point of
passing through some line of
0,
]1
)1,
in
m.
In the finite case
S
(B )
)1
(B )
repre-
\..l
(other than
contains
q
2
~)
aoT-
20
points corresponding to the
L,
2
q
lines of
~
the point of intersection of
subsets of
(B),
which do not pass through
~.
and
We shall now define certain
called sub1ines, which will be the 'lines' of our
~
subplane.
A sub line of
(B
is defined to be a subset of aoT-points common to
(B)
W
) and the set of a.oT-points incident with an aoT-line, provided that the
w
subset contains at least two elements.
to
IT,
are incident with a unique aoT-1ine, they determine a unique sub-
(B)
u
line of
and
Since any two aoT-points belonging
(B).
. w
i
i
We recall that
~
meets
in
L.
is the line of intersection of
Let
and
and
be two lines of
We
have to consider two cases:
Case I.
02
Let
and
meet in point
Q
not on
are the planes representing the aoT-points corresponding to
then they contain the lines as
and
If
m
1
and
and therefore the point
and
Q.
Hence the unique aoT-1ine incident with the aoT-points represented by
01
and
02'
is represented by the point
Q.
Denote by
(~Q)
the set of
aoT-poi.nts incident with the aoT-line of type (1) represented by the point
Q.
Then the sub line determined by the aoT-points corresponding to
(iQ) = (B~) n (~Q)'
is the set
and
whose corresponding lines pass through
Q of
Hence to each point
subline of
(B),
w
Q.
~
,
be the line of
1-1
determine the same sub1ine.
there corresponds a unique
These sub1ines will be said to be of the first
this is the number of points on
Let
i,
not on
In the finite case, there are
Case II.
Q
(B )
which is a subset of the aoT-po:ints incident with the aoT-
line represented by
type.
~,
Any two aoT-points of
m
1
m
1
S
and
(~ ':f~)
m
2
,
~,
q
2
sub1ines of the first type, since
not belonging to
meet at a point
passing through
S'
8' •
l.
on
i (8 ' :f L) • Let
Then the aoT-point
21
corresponding to
.6'
ing
and
m.
m..
is represented by the plane
contain-
O.
1
Hence the unique aoT-line incident with the aoT-points
1
represented by
= 1,2)
(i
1
01
and
02
.6'.
is represented by
(~.6')
Denote by
the
.6' ,
set of aoT-points incident with the aoT-line of type (2) represented by
then the sub line determined by the aoT-points corresponding to
m
2
is the set
<£.S ' ) =
(B ) n (~.6').
J.l
ence between
S'
and
pass through
S'
determine the same subline.
on
u
.e
(S'~L),
S
Any two aoT-points whose corresponding lines
S'.
passing through
S'
Hence to each point
In the finite case, their number is
of intersection of
J.l
The set of points
.6),
and
(B ),
J.l
aoT-line of which
or
of
J.l
q.
L,
the point
together with the set of sub lines of
(.es ')
aoT.
a~T-point
t,
or a point
or
(B),
of
(t ').
s
Il
(m
(B )
Il
S'
on
(B ),
J.l
It is clear that the
is a subset is incident in
(.e )
Q
or
(.e
aoT
with
s ')
con-
Hence incidence in the subplane can be naturally de-
fined by the containing contained relation, and is carried over from
m of
,
Taking the two
(other than
if and only if the subline
tains this aoT-point.
the
.6 '
there corresponds a unique subline.
constitutes a subplane of the affine plane
The line
of
These sublines will be said to be of
cases together, we see that to each point of
a~T-point
correspond-
which is a subset of the aoT-line represented by
the second type.
an
and
there corresponds a unique subline of the subplane
represented by
the line of
.6 ' •
(1,1)
Note there is a
m
l
not passing through
L)
may be called the image of
to which it corresponds, and the point
t
(S'#L)
Then a subline of
aoT.
Q not on
may be called the image of the subline
(B )
Il
is incident with an aoT-point of
if and only if the point which is the image of the subline is con-
Il
tained in the line which is the image of the aoT-point.
22
4.
Two sublines of
element in common.
may be defined to be parallel if they have no
(B)
IJ
We shall show that the parallelism in
over to the subplane represented by
is carried
a~T
There are three cases to be con-
IJ.
sidered,
Case I.
and
Let
be two points of
Q2
are images of sublines
Q2
the line
meets
IJ
at a point
.6 ,
then
Q
l
of the first type.
If
Q Q2
l
and
(£-Ql)
Then
is the image of a
If, however,
(lQ2) •
(lQ ) have no common
2
However, in this case, the plane p contain-
L,
and
(lQ )
1
aliT-point and are parallel.
ing
QQ
l 2
and
represents a oT-point of the derivation set, so that
.6
the aoT-lines represented by
Q
l
whose vertex is represented by
Case II.
of
m
L
not on
IJ,
(lQ2)
then
.6 ,
common to
(B )
and
(lQl)
does not intersect
QQ ,
l 2
unique aoT-point of
QQ
l 2
and
Ql
IJ,
l
on
Let
They are not
S'
belong to the parallel pencil
not on
IJ,
land
S'
Q is the image of a subline
is the image of a subline
(is')
be a point
(lQ)
Case III.
different from
of the
of the second type.
parallel, and contain the unique aoT-point of
image is the line
(lSd.
Then
Q
2
p.
Q be a point of
(S'IL).
first type and
and
(B ),
whose
i..l
S'Q.
Let
L.
S'
and
Then
S'
1
S2
and
1
be two points of
S'
2
both on
IJ
l,
but
are images of sub lines
and
These sublines do not have any common aliT-point, and are parallel,
1
since the line of
joining
of an aliT-point.
.6 '
1
and
and
S'
2
l,
is
which is not the image
However, in this case, the 3-space
point of the derivation set.
lines
S'
1
.6 '
oT-lines of which
2
of
(lsi)
s,
Consider the
passing through
and
(lS2)
oT
represents a oT-
lines represented by the
S'
1
and
are subsets.
parallel pencil whose vertex is represented by
~
6.
These are the
They belong to the
23
We have now shown that any two sublines of
(B)
which are parallel
).l
are subsets of points incident with aoT-lines which are themselves parallel
being incident with a aT-point of the derivation set.
subplane
(B~)
(B)
u
represented by
).l
Hence the affine
can be completed to a projective plane
by adjoining the aT-points of the derivation set, and a new subline
consisting of the points of the derivation set.
at infinity in
This sub line (the subline
(B )] can be appropriately denoted by
and is a sub-
u
set of the aT-points incident with the aT-line
the line at infinity in
aT,
to obtain
It is readily verified that
which was chosen as
~,
aoT.
is a Baer subplane of
each aT-line is incident with at least one
aT
aT,
point of
(BO)
oT-point is incident with a aT-line intersecting
i.e.,
and each
in a subline.
].l
In
the finite case, this follows directly from the fact that the order of
is
q
2
and the order of
is
q.
aT
In the infinite case, a geometrical
argument is necessary.
5.
aoT.
3.
We are now ready to obtain a new affine plane from the affine plane
This will turn out to be the same as the affine plane
Let the points of
a6
be the same as the points of
points are represented by planes of
~,
other than
lines of
a6T,
other than
~
~
aoT.
of Section
Thus
passing through lines of
L.
a6-
S
We, however, delete all
which belong to any parallel pencil whose vertex is a oTThus the deleted lines are represented by
points contained in some plane
contained in
4
and which are not contained in
point of the derivation set.
(i)
8
a6
.
since there are
,
p
i.e. , all points of
of
~
~
not in
In the finite case, there are
q
3
points in
~
passing through
II;
q3+ q 2
~
lines of
(ii)
and not
S
deleted aoT-lines,
other than points of
II,
and there
24
are
q
2
lines of
S,
other than
a~
deleted are also lines of
~.
The lines of
aoT
which are not
and are called lines of type (1), incidence
a~.
between undeleted aoT-lines and aoT-points being carried over to
deleted
of
~
a~T-lines
The
are replaced by the subplanes represented by the planes
not passing through
~,
which are now called
a~-lines
of type (2).
An a~-point is incident with a line of type (2) represented by
corresponding aaT-point belongs to the set
(B).
~
~
if the
It is now readily seen
that we have obtained the same representation of points and lines, and the
same definition for incidence for
a~
as was given in Section 3.
In the finite case, the procedure by which we have obtained
aoT
a~
from
has been called derivation by Albert [1] and Ostrom [10], [12], [5,
p.223].
They have shown that if a finite projective plane of order
made affine by choosing a line at infinity
a set (called the derivation set) of
q+1
~,
and if in
~
q
2
is
we can find
points
such
that there exists a set of affine Baer subp1anes (whose projective completion
contains all the points of the derivation set), and which have the further
property that any two affine points collinear with a point of the derivation set are contained in exactly one Baer subp1ane of the set then we
can obtain a new affine plane, by deleting the
q3+ q 2
affine lines passing
through the points of the derivation set and replacing them by the Baer
subplanes.
The original plane can in this case be called derivable and the
new plane is called the derived of this plane.
us are Baer subplanes of
aoT.
The subplanes defined by
We have already shown that the projective
completion of any affine Baer subp1ane represented by a plane
contains the aT-points of the derivation set defined by us.
that the affine plane
oT
a~
~
of
~,
Thus to show
is the derived of the dual translation palne
(with respect to the given derivation set) we have only to show that
25
given any two aoT-points collinear in
oT,
with a oT-point of the deriva-
tion set, they are contained in a unique Baer subplane.
This means that
we have to show that any two aoT-points belonging to a deleted a6T-line
are contained in a unique Baer subplane of our set.
Case I.
Let the deleted a6T-line be of the type (1).
Q of
represented by a point
~
Then it is
Q is not a point of
(where
~).
Any
two a6T-pcints incident with this aoT-line are represented by planes
and
02
(not contained in
(other than
~)
E) passing through different lines of
Q.
and intersecting in
01
S
We have proved in Section 3, Lemma
(1), Case II (b), that there is exactly one a6-line of type (2) incident
with the a6-points represented by
01, 02'
This is precisely equivalent to
saying that any two aoT-points incident with an aoT-line of type (1) rep-
Q of
resented by a point
~,
are contained in a unique Baer subplane of
the set considered.
Case II.
Let the deleted
sented by a line
~'
of
S
aoT
(~'f~).
aoT-line are represented by planes 01
not contained in
L).
there is exactly one
resented by
o!, 02'
line be of type (2), and be repreAny two aoT-points incident with this
and
passing through
02
~
(and
We have proved in Section 3, Lemma (1), Case I, that
a~-line
of type (2), incident with the
a~-points
rep-
This is precisely equivalent to saying that any two
a6T-points incident with an a6T-line of type (2), are contained in a unique
Baer subplane of the set considered.
Thus in the finite case, the proof that
a6
have been accomplished by proving that the sets
Baer subplanes in
a6T,
is an affine plane could
(B )
~
represented affine
together with an appeal to the theorem of Albert
and Ostrom, coupled with a rephrasing of Lemma (1) of Section 3.
We, how-
ever, chose to give a direct proof in Section 3, in order to include the
26
infinite case.
between
aaT
6.
It is clear that even in the infinite case the relation
and
a~
is quite analogous to that in the finite case.
We conclude this section with some final remarks.
noted that the translation plane
T
It should be
which forms the starting point of our
investigation is restricted to those translation planes which can be represented in a 4-dimensional projective space by the Bose-Bruck construction
given in [2).
These are exactly the planes for which there exists a coordi-
natizing right Veblen-Wedderburn system, which has dimension 2 over some
skew-field
F
contained in the left-operator skew-field of
planes have been extensively studied by Bruck in [4].
translation planes of degree 2.
spread
S in
E
R.
These
We shall call them
Note that there is no restriction on the
to which the translation plane
T
is associated,
Thus
the spread may be subregular in the sense considered by Bruck in [4], but
it need not be.
namesof
For example, it may be the spread associated with the
Se~,Tits,
LUneburg
regular [4], [9], [14],
and Suzuki
which is far from being sub-
Again it may be a Foulser type spread [6], [11].
The dual of a translation plane of degree 2 may be called a dual translation
plane of degree 2.
In the translation plane
T
(E,~)
is
T, there is a special line
uniquely determined except when
when
E
is arbitrary).
representation of
T,
containing the spread
represent T-lines.
such that
aT
~
transitive for any point
is
If
T
our representation of
aT,
on
E. (The line
E
is
is of degree 2, then in the Bose-Bruck
E,
is represented by the 3-space
E
through the lines of which pass planes which
In the dual plane
(E,~)
T
such that
T is a Moufang plane or is Desarguesian,
the T-line
S,
of
E,
aT,
E
becomes a special point
transitive for any line
the aT-point
E
~
containing
E.
In
is represented by the 3-space
27
~,
and the line
oT-point
~.
represents a particular line of
~
The planes
and
PI
P2
through
~,
oT
incident with the
~
determining
resent two arbitrary oT-points collinear with the oT-point
~.
rep-
We have thus
proved the following:
Theorem 1.
derivable.
If
for any line
is a point of
L:
incident with
~
of
and
(~
Every finite dual translation plane
oT
oT
such that
then
~,
~
oT
oT
of degree 2 is
(~,~)
is
transitive
together with any two points
collinear with it, can be embedded in a derivation set
is a unique point, except when
oT
is a Desarguesian plane).
A procedure analogous to derivation is still applicable when
an infinite dual translation plane of degree 2.
such that
oT
is
(L:
,~
Given any two points
of points
)
PI
transitive for any line
~
and
~,
P2
(0) , collinear with
that a new affine plane
Let
a!:l
collinear with
and including
~
~
,
oT
is
be a point of
~
incident wi th
oT,
L
we can find a set
PI
and
02,
such
can be ob tained by deleting the lines of the
parallel pencils with vertices belonging to
(p),
and replacing them by
new lines corresponding to certain affine Baer subplanes.
completed to a projective plane (L:
a!:l
then can be
is a unique point except when
oT
is
a Moufang plane or is Desarguesian).
The choice of
plane.
PI
and
02
will influence the structure of the derived
Thus it should be possible to derive non-isomorphic planes from the
same dual affine paIne of degree 2.
28
5.
SEQUENCES OF PLANES OBTAINABLE FRavl TRANSLATION PLANES
2,
OF DEGREE
BY DlJALIZATIONAND DERIVATION
We shall now show that the process of dualization and derivation by
1,
~
which we obtained the plane
S,
of the spread
from
T
~
can be continued if the line
S.
belongs to a regulus all of whose lines are in
This gives the geometrical representation of the analogous sequences of
planes obtained by Fryxell [7] and Johnson [8].
Although our procedure is
applicable in the infinite case, we shall confine ourselves only to the
finite case, in the first instance.
2.
~-plane
Let us start with the
ize it and obtain a new plane
o~
points of
o~
that
o~
and the points of
constructed in Section 3.
We can dual-.
such that the lines of
are the
~
are the lines of
is derivable if there exists in the spread
~
such that
R.
belongs to
to
R,
~.
(i
the lines of
R*
1
=
O~.
We shall show
S,
R
a regulus
R consists of q+l lines which are one set
of ruling lines of a hyperbolic quadric.
be denoted by
~
l,2, ... ,q).
Let the lines of
Let
R*
R other than
be the regulus conjugate
being the second set of ruling lines of the
quadric under consideration.
passes through
~,
it is a tangent
plane to the quadric, and contains exactly one line of
R*
say
~
meeting
=
at the point
l,2, ... ,q).
O.
(j
by
S. ,
the intersection of
.
and
t.
~
1
1
J
by
P .•
1J
(i,j
We shall choose the
finity in
Of':,.
IT
Let the other lines of
R*
We shall denote the intersection of
t.
J
Since
=
Of':,
t.
J
and
~
by
T. ,
J
t,
be denoted by
~.
1
and
t
and the intersection of
l,2, ... ,q).
line represented by
Then the points at infinity in
Of':,
0,
as the line at in-
are represented by
29
(i)
the q2
and
(ii) the line
other than
planes through
IT,
~.
0
contained in
Let
be the planes of
passing through the line
derivation set the
8~-points
the planes
. .. ,
fl
q
t.
into an affine plane
line represented by
0,
and the
(R),
ber being
8~-points
by deleting the
a8~
incident with it.
8
not contained in
4
~
or
0,
q
3
in number.
Again
a8~-lines
and not contained in
their number being
~
are represented by planes through
q
4
~,
their num~,
Lines of type (3) are represented by the
other than
O.
Now we can divide the
ao~-lines
S
other than
Lines of type (2)
not contained in
in number.
ao~-lines
a8~-
are of three types.
Lines of type (1) are represented by planes through lines of
manner.
Then
8~-
Points of type (2) are represented by planes of
not passing through
and
Points of type (1) are represented by points of
i.e., points of
4 3
q -q.
~
•
8~
points are of two types.
~,
We shall choose as our
at infinity represented by the line
We can convert
the set
~ but not containing ~
~
q
or
~,
2
q-q
points of
~
into parallel pencils in the usual
belonging to parallel pencils whose vertices are
8~-
points of the derivation set will be deleted and replaced by certain
(affine) Baer subplanes of
each consisting of
3.
q
2
ao~.
ao~-lines
Thus there are
q+l
deleted pencils
.
In what follows, we shall consider three different types of (affine)
Baer subplanes, which replace the deleted lines.
We shall first define
these subplanes and prove the appropriateness of the designation subplane
later.
30
Let
(a)
be a plane through
8*
is not contained in
Hence
11 •
tained in
q
of
2
Then
a8~-points
since it contains only the point
rl,
intersects
8*
in a line
rl
6*
sented by planes of
are
q
3
2
T. ,
J
2
q -q
a8~-points
Let
A*
t.
J
or
m
and
passing through
rl
planes
(ii) the
m
~.
not contained in
A*
be a plane through
2
q -q
there pass
2
q -q
Let
be a point of
8.
~
t,
A*
repre-
Through
1 ::; j ::; q,
8ince
~
not contained in
there
or
rl.
a8~-points
The set of
not belonging to
O.
2
q ,
~
t.
Since
or
there
rl,
Then
repre-
8.
~
The set of a8~-points belonging to it
consists of the
~
rl
other than
t,
(B S .) •
sented by planes of
S.
passing through
~
q2
a8~-points
but not through
O.
repreThere
subplanes of type (c).
Let
(B)
a
be the set of points belonging to a subplane of type (a),
a8~
(b) or (c) of
v
O.
8*
subplanes of type (b) .
may be denoted by
(8~)
The set
a8~-points
q
planes not belonging to
sents a Baer subplane of type (c).
q
is not con-
consists of the
represented by points of
t
through
are
m
but not through
represents a Baer subplane of type (b).
a8~-points
( c)
11.
of
represented by the points of
belonging to this may be denoted by
are
where
8*
subplanes of type (a).
(b)
Then
q
through
T.
J
represents a Baer subplane of type (a) .
not contained in the lines
there pass
m
Then
L:.
belonging to this subplane may be denoted by
consis ts of the
t.
J
not contained in
t.
J
represented by the element
a
of
S4'
Also let
be the set of points incident with the a8~-line of type (1), (2),
or (3), represented by the element
is defined to be the sub line of
resented by
v,
v
(B)
a
of
8 ,
4
Then the subset
corresponding to the
(B
a
)n(8~
a8~-line
provided that the subset contains at least two
v
)
rep-
a8~-points.
31
ao~-points
Any two
of
are incident with a unique
(B )
a
sented by an element
v
of
S4'
ao~-line,
repre-
and hence determine the unique subline
(B )n(M ).
a
v
Two sub lines of
point in common.
will be defined to be parallel if they have no
(B )n(o~v)
a
1
Consider two sublines
corresponding to the
S4'
(B)
a
ao~-lines
represented by the elements
This will show that
gether with the sub lines is an affine subplane of
order
4.
q
2
and
is of order
(B )
a
q,
(B )
a
In the notation of paragraph 3, let
~)
contained in
sublines of
(B *)'
o
must intersect
Let
v
8*
f.
in a line
and
M,
at the point
t.
J
there passes a unique line
must be the plane joining
MS ..
1
Let
y
0*
ao~-
(B8*)n(8~v)
1
y
8*
of
(B *).
~
f
meets
1J
and
M,
and must intersect
T.MS.
J
containing
m
in
belongs to both
(B *)
8
and
1.
and
1.
v
f,
other than
p ..
1J
Thus
v
The subline
y,
P..
at the
The aM-
S .•
(8~).
m,
in the line
~
consists of the a8~-point represented by
a8~-points represented by points of
m
S ..
1.
v
in the line
Through
which meets
t
of type
Then
6
meets
Then
-6.
(not
t.
J
ao~-line
(MIT. , P,,=FT.).
1J J
J
p .. ,
is of
We shall study the
S,
1
-6 •
ao~-points
of
to-
aM
be a plane through
of
-6.
be the plane
point represented by
(f) =
of
must be a Baer subplane.
Recall that
T.
J
(B )
a
Since
be a plane representing an
where m and t, intersect at the point
J
point
aM.
representing a subplane of type (a).
(1), which is incident with at least two
q-l
v2
Otherwise, they intersect in the unique aoT-point in-
cident with the corresponding aoT-lines.
=
and
VI
We shall show that the sublines are parallel if the corresponding
lines are parallel.
d
(B )n(o~v )
a
2
and
and the
and
M.
1.J
This subline may be said to be of type (al) , and may be appropriately represented by
of
0* ,
L
(B *)
8
has
not passing through
q
2
sublines of this type, represented by lines
T .•
J
32
Again let
A
is incident with at least two
in a line
k.
a6~-points
A passes through
Since
which is the only point common to
consists of the
of
k,
a6~-line
be a plane representing an
other than
T ..
of
(B *).
6
~,
k
ao~-points
q
~
.
or
J
ao~-line
of
ao~-points
k.
through
Q
consists of the
m,
line of
ao~-points
q
not passing through
of type (a3).
this must be the unique
(B *)
o
A
represents the
Consider two sublines of
in
i'
and
meeting
ao~-lines
the plane
M'.
t
Let
~.
1
at the point
corresponding to
y
i
and
q-l
subother
T. ,
J
containing
i
O
represented by planes
ao~-point
1J
Let
i
and
m and
T .•
J
in common, then
ao~-
t .•
J
i '.
1
of
6*,
not on
Let
i
and
S passing through
and
S.
A
or
t.
J
m,
common to the two sublines.
of
P ..
S .•
and meeting
J
of type (al), represented by lines
be the line of
1
The subline
T ••
both of type (al), or one of type (al)
(B *)
o
Next consider two sub lines of
lines
has
incident with both the corresponding
ao~-point
meeting at the point
M and
points
Thus there is only one sub-
O.
and the other of type (a2), meet at a point
i
q
It is represented by the point
ao~-point
If two sublines of
then clearly
(k)
passing through
6*
It is clear that if two sublines have an
lines.
(B *)
o
represented by
ao~-line
j
of
The subline
of type (3), which is incident with at least
is the
(Bo*)n(o~T)
(m)
in the point
m.
Finally, the only
two
0*
This subline may be said to be of the type (a2),
J
lines of this type represented by lines of
t.
~
represented by the
and may appropriately be represented by the line
than
A must meet
Then
must meet
and
0*
of type (2), which
0I
i'
meet
m
and
P ..
1J
be the planes representing
Then the
ao~-point
represented by
belongs to both the sublines.
(B 0*)
of type (al) , represented by the
m at the point
M.
Let
i
and
i O meet
33
t.
J
at the points
passing through
Sk
and
P ..
1J
P ..
1J
respectively.
and
Let
~i
and
and
y
y'
joining
m with
Thus the sublines represented by
are represented by the planes
8
and
s,
be the line of
t
at
S.
land
S.
1
and
1
Then the a86-points of type (2) belonging to
point in common, and are therefore parallel,
spectively.
~k
respectively and meeting
P .
kJ
represented by the planes
spectively.
Pkj ,
lO are
land
and
Sk
re-
lo have no a86-
The corresponding a86-lines
joining
M to
~k
and
re-
The a86-lines are parallel, and meet the line at infinity in
the 86-point represented by the plane
SiSkM
(which is an element of the
derivation set).
l (defined
Consider now the intersection of the sub line represented by
as before) and the sub line of type (a3) represented by the point
plane
y
= MS.T.
1
joining
J
m and
S.,
1
The
T ••
J
represents an a86-point common to
both the sublines.
Finally it is readily seen that two sub lines both of type (a2) or one
of type (a2) and the other of type (a3) are parallel.
The corresponding
a86-lines are also parallel meeting the line at infinite in the 86-point
represented by
~
(which is an element of the derivation set).
This completes the proof that
an affine paIne of order
a86.
q,
(B *)
8
tegether with its sublines is
with the same incidence and parallelism as
Hence it is a (affine) Baer subplane of
a86.
In the same way, we can prove that the sets
(B *)
A
and
(B S )
defined
i
in paragraph 3, together with their sublines are (affine) Baer subplanes of
a86.
5.
Details of the proof will be omitted.
In the previous two paragraphs, we have defined a set of
Baer subplanes of
a86,
q3
of type (a),
2
q -q
q3+q 2
of type (b) and
q
affine
of
34
type (c).
We shall now show:
Lemma (7).
Any two points of a deleted line (i.e., an a06-line which
meets the line at infinity in a point of the derivation set) are contained
in a unique Baer subplane of the above set.
The 06-points at infinity belonging to the derivation set are represented by the line
~
~
meeting
and
~.
S
~
planes of
passing through
t.
Let
An a06-line, the point at infinity on which is
d.
Let
d
passing through
and
S .•
1
It is represented by a plane
t
intersect in
Thus
Si
0
~i
and let
is the plane containing
~.,
1
(i)
Consider two a06-points of type (1), incident with the a06-line
represented by
O.
Let them be represented by the points
1 =
Suppose first that the line
other than
S .•
1
Through
R* , and meeting
and
q
must be of type (1).
in a line
be the line of
O.
and the
be one of these planes.
represented by
d
~.
~
pp'
meets
at the point
Then the plane
It is of type (a).
passes through
S .•
1
(B *)
A
J
0*
of
pI
p .. ,
1J
belonging to
t.
containing
t.
J
which contains the given
1 =
Next suppose that the line
Then the plane
the unique Baer subplane
(B *)
o
and
at the point
~.
1
there passes a ruling line
P ..
1J
1 represents the unique Baer subplane
a06-points.
P
1 and t
containing
containing the given
pp'
represents
ao~-points.
It is of
type (b).
(ii)
Consider two ao6-points, one of type (1) and the other of type
(2) incident with the ao6-line represented by the plane
represented by the point
P
the line of intersection of
of
0,
0
and
and the plane
~,
d
y
O.
of
is contained in
Let them be
~.
Since
y.
Hence
d
is
y
35
meets
IT
in a line
T,
J
Through
where
T,
is a point of
there passes a line
t,
of
the line joining
Then
subplane
and
P
Po,
~J
a8~-line
J
J
meet
d
t,
and
J
in
m
a8~-points
Next consider two
represented by the plane
y
R*
~,
meeting
M.
Let
y',
and
in the lines
8.
other than
~,
~
m
in
J
1
T,M.
J
Baer
It is of type (a),
Let
them be represented by the
Then
(B S ,)'
Let
•
both of type (2) incident with the
and
the unique Baer subplane
~J
O.
be the line
which must both pass through the line
S.T,
P"
represents the unique
which contains the given a8~-points.
(B *)
8
(iii)
IT
J
~
the plane containing
8*
planes
SoT,
which contains the given
d,
and meet
S.
represents
1
a8~-points.
It
~
is of type (c).
a8~-lines
Let us now consider
~.
resented by
(iv)
A
Such a line may be of type (2) or type (3).
A be a plane representing an a8~-line of type (2).
Let
~.
contains the line
Let
PI
to consider two separate cases.
meets
~
for which the point at infinity is rep-
T,
J
at a point
and
2
be two points of
o.
Let
t,
J
T, •
P P .
l 2
represents the unique Baer subplane
the
Then
8*
a8~-points
represented by
pose that the line
t
passes through
P P .
l 2
Then
(B *)
A
P P
l 2
O.
T.
J
~
Let
T,
J
is a point of
representing
P
l
and
P .
2
meets the line
Let
\*
and
be the plane containing the lines
(B *)
8
which contains
It is of type (a).
~
at the point
O.
Next supRecall that
be the plane containing the lines
t
P
l
and
P .
2
It is of type (b).
be a point representing an
~,
other than
a8~-points
and
a8~­
is the unique Baer subplane which contains the
points represented by
(v)
8*
We have
be the line of
passing through
J
A.
Let us first suppose that the line
other than
Let
P
Then
O.
Let
Yl
a8~-line
and
of type (3).
Y2
of type (2) incident with the
Then
be two planes of
a8~-line
represented
36
by
Then
T .•
J
m the line of intersection of
Also let
T .•
J
J
the plane containing
subplane
and
t.
J
m.
passing through
Then
8*
Y2
passes through
T ..
Let
J
be
8*
represents the unique Baer
containing the aM-points represented by
(B 8*)
is of type (a) .
b.
R*
be the line of
t.
and
Y1
and
Y1
Y2 •
It
This completes the proof of Lemma (7).
We can now use the theorem of Albert and Ostrom already referred
to in Section 4, paragraph 5, to derive a new affine plane from
can then be completed to a projective plane.
a8~,
which
Before proceeding to do
this,~
shall make two observations necessary for elucidating the structure of
the new plane.
S-R,
be the set of lines obtained by deleting the lines
belonging to the regulus
So u R*.
Then
S*
R from the lines of the spread
is a spread since the lines of
exactly the same points.
placed by
t,tl, •.. ,t
The lines
1
of
S
R*
Let
=
S*
contain
merely get re-
S*.
to give
q
-6 ,-6 , ... ,-6 q
Rand
S.
Note that for an affine Baer subplane of type (a) or (b), represented
by the plane
0*
contained in
(B *)
o
0*
or
y*.
or
or
Again an
is contained in
0*
A* ,
or
A*
(B o *)
an
a8~-point
(B *)
A
if and only if
a8~-point
P
P is
is a point of the plane
of type (2) represented by the plane
(B *)
A
or
of type (1), represented by
if and only if
y
y
intersects the plane
in a line.
Similarly for an affine Baer subplane of type (c) represented by the
point of
a8~-point
(B S . )
~
S. ,
~
the set
(B S . )
contains
a8~-points
of type (2) only.
~
of type (2) represented by the plane
if and only if the point
S.
~
y
is contained in the
is contained in the plane
y.
An
37
We now apply the procedure of Albert and Ostrom to the affine plane
a06.
The points of
a06
are retained but the
q3+q2
lines of
a06
which
belong to parallel pencils, having one of the 06-points of the derivation
set vertex, are deleted and replaced by the
obtained in paragraphs 4 and 50
q3+q2
affine Baer subplanes
It follows from Lemma (7) and Ostrom's
theorem, that we obtain a new affine plane which we may appropriately denote by
a06*
as
bears to
a06,
a06
since it bears exactly the same relation to the spread
S.
The points of
a06*
and the lines are of three types:
S* ,
by planes through lines of
their number being
through
t
4
q.
S*
are the same as the points of
Lines of type (1) are represented
other than
t,
and not contained in
z:; ,
Lines of type (2) are represented by planes
not contained in
(3) are represented by the
or
~
q
2
L,
q -q
points of
t
in number.
Lines of type
o.
other than
Incidence is
always given by the containing contained relation except between a06*points of type (2), and a06*-lines of type (2) or (3), when incidence means
that the representative planes intersect in a line.
We can now complete
infinity in
the
q+l
the line
06*
ao~*
to a projective plane
o~*.
is obtained from the line at infinity in
The line at
06
by replacing
06-points of the derivation set by new 06*-points represented by
t,
we denote by
and the
(D*)
q
planes of
~
which contain
~,
but not
t.
If
the set of new 06*-points, and incidence is defined as
before, i.eo, by the containing contained relation except in the case when
a 06*-point and a 06*-line are both represented by planes, when incidence
means that the representative planes intersect in a line, then it is readily
verified that the element of
8
4
representing any affine Baer subplane of
our set is incident with exactly one element of
(D*).
The affine Baer
subplane can therefore be completed to a projective line, by adding this
38
element of
(D*).
ao~
Any undeleted line of
ao~*
is a line of
also.
It was obtained in the first place by deleting a point at infinity from a
o~
corresponding
line.
get a projective plane
This is now again added to it.
66*~
7.
6~*
We can now dualize
o~
which is the derived of
actly the same relation to the spread
S*
as the plane
~*~
to obtain a plane
In this
way~
we
and bears ex66
S.
does to
where the lines of
the one are points of the other and vice versa.
TIl
Since derivation is an involuntary process (i.e., if
of
TI2~
TI2
then
lation to the spread
of Section
oT*~
T*
4~
S*
as
derive from
~*
~
~*
TIl) and since
is a derived of
to
S~
we can by reversing the procedure
Finally~
T*.
oT*.
Dualizing
we can derive
T
from
by using the 'switching' process of Bruck and Bose [2]~ which is equiv-
alent to derivation in which the line at infinity in
by
bears the same re-
a dual translation plane
we obtain the translation plane
is a derived
E~
the points at infinity by the lines of
S*,
T*
is represented
and the derivation set
We thus get a cycle of eight planes~
T~ oT, ~~ o~~
by the lines of
R*.
o~*~ ~~
the relation between which can be schematically repre-
6T*~
T*
sented as
o
T
D
~
D
oT
D+
t
D
T*
where
o
~
stands for
+
dualization~
oT*
+
and
~
D
~*
+
o~*
for derivation.
direction of the arrows could have been reversed.
Of course the
39
Finally we note that even in the infinite case we can obtain
from
o~
by a procedure analogous to derivation, under the assumption
S,
that the spread
belong to
o~*
R.
R,
contains a regulus
and the line
~
is chosen to
We can thus enunciate the following theorem (compare
with
Fryxell [7] and Johnson [S]).
Theorem 20
Given a translation plane
= TI , which can be linearly
T
represented in 4-dimensional projective space by the Bose-Bruck construction
S
of Section 2, and if the spread
then we can obtain a cyclic sequence
planes, such that
the derived of
TI
TI
Zi
that each plane of
i
'
= TI i +S '
TI
2i
defined there contains a regulus
{TI. l,
~
i
= 1,2,3,coo
is the dual of
TI 2i - 1 ,
R
of projective
and
TI Zi+1
is
provided that in the infinite case we further assume
L
contains at least one line of
So
ACKNOWLEDGfvENT
The authors acknowledge with pleasure the helpful discussions with
Mrs. J.G.D.S. Yaqub of Ohio State University.
40
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1.
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"The finite planes of Ostrom".
(1966), 1-13.
Matematica Mexicana,
11
Boletin de la Sociedad
2.
Bruck, R.H. and Bose, R.C. "The construction of translation planes
from projective spaces"c J. Algebra, 1 (1964),85-102.
3.
Bruck, R.H. and Bose, R.C. "Linear representations of projective
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4.
Bruck, R.H.
"Construction problems of finite projective planes".
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Chapter 27 (1969), 416-514.
Finite Geometries.
5.
Dembowski, P.
1968.
6.
Fou1ser, D. "Solvable flag transitive groups".
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7.
Fryxe11, R. "Sequences of planes constructed from nearfie1d planes
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8.
Johnson, N.L.
Soc.,
9.
16
Springer Verlag, New York Inc.
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Math. Z.,
86
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Notices, Am. Math.
LUneburg, H.
"Die Suzukigruppen und ihre Geometrien."
Springer Verlag, Ber1inGotingen-Heide1burg-New York, 1965.
Lecture notes in Mathematics, 10.
10.
Ostrom, T.G. "Semi-translation planes".
1ll (1964), 1-18.
11.
Ostrom, T .G.
Trans. Amer. Math. Soc.,
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Combinatorial
Mathematics and its Applications. University of North Carolina
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Tits, J. "Ovoides et groupes de Suzuki".
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