•
This research was supported in part by the National Science
Foundation under Grant GP-23520 and by the Air Force Office of Scientific Research under Contract AFOSR-68-1415C.
LINEAR REPRESENTATIONS OF DERIVED SHEARS
PlANes·
by
Nicholas Krier
Department of Statistios
University of North CaroUna at Chapel HilZ
Institute of Statistics Mimeo Series No. 760
June, 1971
LINEAR REPRESENTATIONS OF DERIVED SHEARS PLANES*
Nicholas Krier
Department of Statistios
University of North CaroZina at ChapeZ BiZZ
ABSTRACT
A linear representation is determined for any semitranslation plane derived from a shears plane
case by Bose and Barlotti.
representation of
oT
oT,
which generalizes that determined in a special
The connection is made between the geometry of the
in the case that
imposed on coordinatizations of
oT
oT
is derivable and the conditions
determined by Ostrom.
Finally, we note
that a family of translation planes discovered by Knuth are derivable.
The
derived planes are not strict semi translation planes but they fall in general
outside the class covered by Bose and Barlotti in their representation.
*
This research was supported in part by the National Science Foundation
under Grant GP-23520 and by the Air Force Office of Scientific Research under
Contract AFOSR-68-l4l5C.
2
§l. PRELIMINARIES
Definition:
tinct elements
A pight quasifield
Q
= (Q,
+, 0)
is a set containing dis-
and admitting 2 binary operations "+"
0, 1
and
"0"
called
addition and multiplication and satisfying
1)
(Q, +)
2)
o
3)
(Q\{O},
4)
(a+b) ° c
5)
for
0
is an abelian group with null element
=a
a
=0
o 0
a
in
Q.
is a loop with identity
0)
=a °
c + b o c
in
r, s, d
x ° r - x
for all
Q,
1.
for all
r :f s
O.
a, b, c
in
there is a unique
Q.
x
in
Q so that
s = d.
0
To each right quasifield, we associate a translation plane as follows:
points are ordered pairs of elements of
{(x,y): x
= C,
CEQ}
or
Q,
lines are sets of the form
{(x,y): y = xom+b}
and incidence is given by
containment.
The kePneZ of a right quasifield is the set of all
ko(x+y)
= kox
+ koy
and
ko(xoy)
= (kox)oy
for all
k
such that
x, y
in
Q.
The kernel is in general a skew field but since all planes and quasifields
in this paper are finite, the kernel will be a field.
If
H is a subfield of the kernel of
of dimension
with
rover
H for some
Let
in
Q write
of
r-l
s
m
= {(x,xom):
S2r
in a member of
not in
S.
S2r
Let
Soo
x in Q}.
Let
S
spaces and these elements of
Q is a left vector space
T associated
dimensional projective space over
H.
S is called an r-l spread.
points of
2r
L be a hyperplane in
as r-vectors with components from
then
The translation plane
Q may be represented in S2r' the
H, as follows:
set
r.
Q,
and write the elements of
= {(O,x): x in Q} and for m
=
{s } u {s :m
00
m
in Q}.
S partition the points of r.
The points of
L and the lines of
Q
S
consists
Such a
T are represented by the
T are those
Incidence is given by containment.
r
spaces meeting
r
3
T may be extended to a projective plane by adjoining
of
r
and the members
S as the line and points at infinity respectively.
T may be described by means of the spread
specific quasifield.
representation of
Such a representation of
S and without reference to a
T will be called the usual
T.
A shears pZane is the dual of a translation plane and may be represented
in
S2r
by interchanging the definition of point and line.
The shears plane
may be given an affine representation by deleting one member
S as the line at infinity and all points on it, namely r
meeting
If
r
in
of the spread
and all
r
spaces
s'.
T is coordinatized by
we require that
=S
Sl
00
Q then the dual of T,
denoted by
oT
may be coordinatized by the left quasifield
)
Q but with
with the same addition as in
plication in
s'
a*b
= boa
where
*
(here
Q*
denotes multi-
Q*.
In a projective plane
2
of order
P
m ,
a subplane of order
m is called
a Baer subplane.
In a projective plane of order
2
a derivation set is a set
m ,
points on some line
W such that for any pair of points
the line joining
to
generated by
If
P
x
x
and
in a point then <Vu{x,y}>,
is made into an affine plane
the lines of
P
not on
6P
~
by deleting
P,
points are the points of
meeting
W in a point not in
The Bose-Barlotti representation:
2
over its kernel.
space over its kernel.
contains the line
Sl
if
the subplane
V,
W,
and
then a new affine
6P
and ii)
lines are
those Baer
subplanes whose points at infinity are precisely those elements in
gree
W,
~l
y, is a Baer subplane.
plane may be defined:
i)
meets V
y
x, y
V of
Let
Represent
n
be a
of the spread
Let
T
T be a translation plane of de-
in the usual way in projective
a space
S.
V.
meeting
r
in a plane
In the shears plane
oT,
IT
4
which
we define
4
the derived plane
boT:
S distinct from Sf,
and
boT
boT
ii) the planes in
lines are
meeting
Q
L in a line of
points are those planes meeting
i) the points of
IT
8
L or
not in
4
in a line distinct from
Sf.
Q,
Incidence
of a point and a line of type i) is given by containment and a point and a line
of type ii) are incident just when the planes representing the point and the
8 ,
4
line meet in a line of
The Baer subp1anes in the derivation of
are merely planes in
planes in
L in
oT
jective space
8
is derivable.
Theorem 1.
An r-space
Baer subp1ane of
Proof:
p
s
Let
R meeting
be a point in
x
The T-line
x.
points not in
dimension
s-I.
has dimension
s'
L
L
so that
= Ze,
r
a prime, such
p
in the usual manner in the pro-
GF(p).
L in an
r-1
space represents a
S
in
s
8-1.
B of
R but not in
= <s,x>
cUm LnR
For each ideal point of
either
Rns
= 13
or
B,
s
Thus
we find an
in
LnRnL
s
does not represent an ideal point of
R will contain
pr
=
B,
then
Ps +1 members of S meeting
(ps)Z
points not in
L.
pS+1
B has
and let
L
for some
= s.
Then
T.
L be a line
S must meet
= Rns
is of
such that
But these intersections exhaust the points of
verse1y, there will be precisely
space.
over
R represent a Baer subplane
B containing
that if
Zr
T
T if and only i f for each
Let
ideal points.
R in
Represent
p ,
= s-l·
cUm Rns
of
of dimension
Zr
and those
E
PLN~ES
r
T be a translation plane of order
that its dual
is the 3 space
s'.
§2. LINEAR REPRESENTATIONS OF DERIVABLE SHEARS
Let
are desarguesian, since they
V
The derivation set
8 ,
4
meeting
Q
oT
Rns'
R
RnL,
= 0.
in an
Rns
so
Cons-l
Each T-line through
5
one of these
pS+l
spread elements containing a point of
R in an s-space and so each line contains
meet
Further for
x, y
in
them and this T-line meets
(4.b»
sents a Baer subplane of
on
r,
then
If
in
B,
=B
Let
=
B mapping
in
B.
is a collineation of
and so
<Eu{x,y}>
a
=
<Eu{x,y}> a
If
x
x, y
R repre-
and so
R.
T
fixing each member of
of
B to points
induces a collineation of
a cr
<Eu{x ,y }>
=
y,
B.
a
a
x , y
S
set-
also
B.
Then
a
a a
<E u{x ,y }>
a
B .
=
B and there is a translation
then the affine points of the line
Such a translation has order
points on it and each of the
p
this translation must also lie in
lie in
s
T having more than one point
x, yare affine points of
to
p
T.
E be the set of ideal points of
Lemma 2.2.
of
a
r.
R not in
points so that by Dembowski (p.l39
B is a Baer subp1ane of
If
a
B
B =
Proof:
s
maps the affine points
a
then
Proof:
p
points of
there is a unique T-line containing
B is represented by an r-space
Lemma 2.1.
wise and if
in
R
s
R lie in an affine plane of order
the points of
Theorem 2.
r,
R but not in
p
r will
R not in
images of
B.
p,
x
the line
<x,y>
<x,y>
has
also lie
p
affine
under repeated application of
Thus all the affine points of
<x,y>
B.
This lemma need not hold if the plane
larger than
GF(p).
for larger fields.
T
We shall see examples in
is represented over some field
§4
where Theorem 2 will not hold
6
Lemma 2.3.
x, y
If to any pair
B mapping
translation of
to
x
of affine points of
y
B there is a
B lie in an
then the affine points of
r-space.
Proof:
Let
x, y
translation mapping
and
E,
is in
z
to
x
y.
let
If
z
x
to
map
2
B
plane
so this plane lies in
steps the group
of order
p
and
= <ol,oZ, ••• ,or>
G
r
Let
G
1
z.
G
2
Then
= <0 1 ,0 2 >
lation of
B
Proof:
y
to
x
Let
x, y
mapping
For
p
B.
and so with
maps
p > 2,
let
E have equation
y:(l,O •••• O.I).
,
eation of
point of
Let
B.
Further
0
x
= y0 ,
0
B.
,
B.
B.
of
Then there is a trans-
x
mapping
l'
(~I"'~2r,n)
We may assume that
E,
exchanges
,
w, w
Let
B.
x
y
to
for
S2r
x:(O,O •••• O,I)
and
with
y
maps
and
and so is a co1lin-
z:(t,O ••• O,I).
fixes the point
is a translation of
duces a translation of
r
Lemma 2.1 gives the desired result.
B not on the T-1ine containing
therefore also in
is a col1in-
be the mapping
fixes all points of
and the T-line through
Then
= x0 ,
= 0.
,
y
y.
introduce homogeneous coordinates
n
to
x
the
1
to each of the points in an
x
the translation
Y
0
Continuing this procedure, after
be 2 affine points of
to
= 2,
For
Then
x
with
is mapped to all points of the affine
x
affine r-space and this exhausts the affine points of
Lerrona 2.4.
= <0 1>
is not on the line joining
eation group of
<x,y,z>,
B.
be affine points of
x
and
y.
Then
Let
w'
meets the T-line through
0
w be an affine
is also in
in
z which is
be the mapping
T mapping
How
02
x
to
z
and
z
to
is the desired translation.
Lemmas 2.4 and 2.3 now yield the conclusion of the theorem.
y
B
so
0
in-
7
Note that in
oT, oil
is a Baer subp1ane containing the
Theorem 2 implies that any Baer subp1ane of
oT
containing
oT
point
E so
E is given by
an r-space,
Let
in S
Soo
be a line of the shears plane
oT
and let
v = {E,A 7",A
} be a derivation set on s00, We assume explicitly that
pS
so that Theorem 2 will apply, The A. are r-spaces meeting E in
is in V
~
s 00 •
Lemma 2.5.
Proof:
Let
The spaces
Let
sl :/: s 00 :/: s2
joining line is
lie in a space n of dimension 3 s.
ps
be members of S. Let q1 be a point in A •
7
L2 ... <s2,Q2>'
L1 ... <sl ,q1>'
derivation set
A7 .. ,A
Then
V in the
Q1 which meets
oT
R.
cUmn ...
r+r-1-(s-1)
oT
Further,
<V,L ,L >
1 2
line of
A ... A
7
ps
Eo Thus
represented by an
r
00
L
1
meets
Ai
in a point
Qi
rep-
and so
...
Thus the spaces
V is a
Since
A7 •
oT,
oT
points whose
cUmRns ... s-l and so
From Theorem 1,
= 2r-S = 3s.
oT
point
<0,L ,L > is a Baer subp1ane of
1 2
space
resenting a
are
L , L
2
1
... n.
c
lie in
n.
Note that the
Ai
parti tion the
} and
ps
7
does not depend on the elements of the spread used in the proof of Lemma 2.5.
n
points of
Let
IT
not in
= nnE.
Now
any
is determined by the set
cUmIT ... 3s-1. For each s:/: s
00
in
so that from Theorem 1 we have that
s :/:
8
00
could have been used so that for each
There are
is:
Then
n
8 in
pr
S, 8:/:S}
00
members of S
distinct from
partition the points of
IT.
{A ,· .. ,A
S,
let
cUms
...
, 1
8-1.
- cUms = 8-1.
s,
s 00
and so
S'" snIT.
sand
00
But
8
Lermna 2.6.
by
i) dim<8 ,8 > .. r-1,
i j
R the space
subp1ane of
8k OR .. ~
then
8i o8 j .. ".
is the intersection with
containing V and the
oT
or else
Proof:
R:
8ke R:
for all
oT
sk ~ soo'
lines
M .. <Sj,q1>·
2
M1 .. <si,q1>'
si
and
n
Sj'
iii)
-si
-Sj
and
and the fact that
Since V is a derivation set
<'D,M ,M > is a Baer subp1ane of oT containing l:, M and M
1 2
1
2
linear subspace R representing this subp1ane contains
<8 ,8 > = Ron = R.
i j
Lemma 2.7.
Soo
The subspaces of
8-1
R rf: R'
If
n
generated by pairs
sk
j
where
R and
R'
R by Lemma 2.6 (iii).
dim<R,8 > ... r-1+s-1-(-1) .. 3s-1.
k
Thus
Thus
partition
R'
dimn ... 3s-1
RoR' .. 8
n
generated by
contains some
~
dim<R,R'> ~
and so
St or else
for some
t
dimRoR' ..
These last equalities follow from the fact that the
dimension of each intersection is
s-l.
it must be shown that each point of
be in
For
dim<R,R'>'" 3s-1
From Lemma 2.6 iii) and iv), either
Ros 00 .. ROR' .. R'os.
00
<si,Sj>
are subspaces of
dimRoR' ... s-l.
m n
disjoint from
s-l.
and so
spaces.
<8 ,8 >, <S ,s > respectively, then
i
and hence the
iii) and iv) now follow from Theorem 1.
into a spread of
Proof:
of a Baer
iv) dimR:os oo ... 8-1.
i) follows from the dimension of
Let
ii) if we denote
To complete the proof of the lemma,
Soo
n but not in S00' and let t
is in some
be the line
R.
Let
<q,q'>.
contain one point of t[8 j contains at most one point of
Let R .. <srsk>· Then R contains l and hence q.
t
qE:s.
00
Let
Let
-Sj' -sk
since sjosoo
III
q'
each
,,].
\s·
We define
to' t 1 ··•
an incidence structure P in n. The P points are {to ... t , 81 , ... ,8 }.
pS
pr
Let the (S-l)-spread in
The P-lines are
s
CD
Soo
contain the spaces
and the subspaces
is given by containment.
where
si
rf:
Sj.
Incidence
9
Theorem 3.
Proof:
s
is a desarguesian projective plane of order
P
There are
s 2 s
(p ) +p +1
P-points.
p •
Each P-1ine contains
pS+1
P-points since each P-1ine is an r-1 space and the P-points on it form an
(8-1) spread on the P-1ine.
To show that 2 P-points are on a unique P-1ine,
we have only to consider the P-points
q!t i , q'!Sj
Let
line
<q,q'>.
2.7,
tic<Sj,sk>
and let
Then
sk ~
-sjc<Sj
- .sk>
-
t , Sj
i
Sj be such that sk contains a point of the
and
q
and this is the only
since
<ti,Sj> .. <Sj,sk>·
s
plane of order p
as the other 2 cases are obvious.
is in
Thus from Lemma
<sjnsk>nt i •
and Sj
i
P is a projective
space containing
r-1
Thus by Dembowski (p.138,3.c)
t
.
In the language of Bruck and Bose [2], this representation of
for every point of
joining
x
to
y
P.
The representation is called rigid at
is represented by the subspace
<x,y>
x
P
to be desarguesian.
if the P-1ine
for every
But from [2] p.134 Corollary to Theorem 6.1, it suffices that
represented for 2 points for
P is rigid
Thus
y
~
x.
P be rigidly
P is desarguesian
and the theorem is proved.
Coro ZZa:PJf :
Each of the Baer subp1anes of
cisely the points of 0,
Proof:
Taking
Let
R .. RnI! .. <si,Sj>.
R represent such a plane and let
of the affine plane
pR.
P, the P-points on
Let
iL Then any P-point not on R meets
$
Further each P-1ine
since dim(R'n<j»" r-1+r-(3s-1) .. s
Thus the spread in
R are just the ideal points
We give a representation of
plane in the usual manner as follows:
SkCR.
having ideal points pre-
is desarguesian.
R as an ideal line in
s-1+r-(3s-1) .. O.
oT
$
pR as a translation
be an r-space in
in a point since
R'
PR meets
of
and this
s
I!
containing
cU.m(skn<j»"
<j>
space meets
in an s-space
R
in a space
R is the spread for a desarguesian plane and does
not depend on which r-space containing
R is used to represent the points.
10
Thus the r-space
subplane of
R itself represents a desarguesian plane which is a Baer
oT.
R
The proof could be facilitated somewhat by arguing that the spread in
p. 2.
is regular, but then the conclusion would not be valid for
Theorem 4:
~-points
from
The derived plane
are the r-spaces of
s~,
(ii) the r
~-lines
~oT
S2r
metting
are (i) the points of
n
spaces in
meeting
IT
line of type (i) is incident with a
point and a
can be described as follows:
~-line
E in a member of S
S2r
in neither
different
E nor
in a P-line different from
~
n and
s •
~
point if it is contained in the
of type (ii) is incident with a
~
A
~
~
point if their repre-
senting r-spaces meet in an a-space.
Theorem 4 is merely a restatement of the general process of derivation in
terms of the representation of
oT.
The description of Theorem 4 will often be possible over a field larger
than
GF(p)
but we shall give some examples in
of the projective space
S2r
§4 where
the underlying field
must then be a proper sub field of the kernel of
T.
Theorem 5.
S2r
over
If
GF(p) ,
dimensional space
T of order
then
ITcE
oT
is represented in the usual manner in
is derivable if and only if there is some
such that
other members of S meet
pr
IT
IT
contains some member
s
~
of S
38-1
and the
in the affine points of a rigidly represented
desarguesian plane.
Proof:
let such a
One direction has been proved in Theorem 3.
IT
exist.
Let
n meet E in
forms a derivation set where the
Ai
IT.
For the converse,
We show that
are T-lines containing
V·
{E,A ••• A }
7
ps
s~
and contained
11
in O.
Suppose
Sl • sl nTI ,
L1 , L2
9 2 • s2 nTI
are 2
oT
and let
R
points whose
= <q,sl,9 2>.
oT
line
q
is in some
Ai.
and
are in S.
Then
R is an r-space and by
Let
the definition of rigidity satisfies the conditions of Theorem 1 so that
represents a Baer subplane of
are
L2
L2nE.
oT
Let
q1· A1nL 1 ,
~
Thus
§3.
sand
q2· A2nL
2
V
a
in
E,
and let M be the
oT lines
<Vn{L 1 ,L 2 }>
<VU{L ,L }>.
1 2
This subp1ane is just
points whose joining oT-1ine meets
the intersection of the
s'
oT.
and
<Vn{L ,M}>
1
then
oT
LInE
If L ,
1
=s
•
point representing
M meets
in a space
so we have returned to the first case.
V is a derivation set.
CooRDINATIZATION OF DERIVABLE SHEARS PLANES
Let
be given the homogeneous coordinates
L
Pl, ••• ,P ' 0l' ••• 'os)
S
Let
with components elements of
are
O.
Let
to' t 1 be P-points on
sm be given by the equations
S
m
nl
(nl,···,n s '
GF(p)
= ••• = nS
~l'
•••
'~S'
not all of which
•
~l
= ••• •
space over
GF(p)
for copies of
where
°
~
(0,0, .•• 0)
= GF(pS)
and so choose
over
each of the
•••
given by
and from
-
82 are of the form
(x,O,x,x)
and the second component in the vector denotes the s-vector
GF(p)
•
{ol ••• oS}' {Pl ••• PS}, and {nl ••• nS} as bases
corresponding to the null element of
F.
The 4 P-points
F(O,O,l,O),
a
may be considered as an s-dimensional vector
F such that the members of
x€F
= O.
and
=
F
~S
given by the equations
• Ps = 0 respectively. Let -8 1 have defining equations ~l
~S • PI = ••. • P = 0 1 = ••• = aS = O. Then TI = <sm,sl> and TI is
s
the equations r; =
• 7;S = O. Let S2 be disjoint from <to,sl>
1
PI
R
t ,t ,sl,s2 form a quadrangle in P with coordinates
o l
F(O,O,O,l), F(l,O,O,O) and F(l,O,l,l) respectively. Thus
have coordinates over
F of the form
F(l,O,m ,m ).
l 2
12
Let
sm
+s
Q be a quasifie1d coordinatizing
{(x,xom): x in Q}.
are given by
~
an ordered pair of elements from
If
0; feF
s
and
sm· F(1,0,m ,m )
1 2
write
m = (m1 ,m2).
(f,0,fm ,fm ). On the
2
1
But the difference of these 2 vectors
then so is
since the first 2 components vanish.
~
Q
We associate to each element of
For
(1,0,m ,m )es ,
1 2
m
«f,0),(f,0)0(m ,m »es •
1 2
m
other hand,
is in
F.
T so that the spread elements
Thus the last 2 components
(1)
Theorem 6.
Identifying
(f,O)
with
f,
for
feF,
F is a subfie1d of
Q and Q is a left vector space of dimension 2 over F.
Proof:
Since
(f,O)o(g,O)
isomorphism between
Now
F and
(£g,O)
{(f,O): feF}
and
(£,O)+(g,O).
~+g,O)
the
is clear.
f o (go(m1 ,m2» = f o (gm1 ,gm2 ) = (fgml,fgm2) • fg o (m ,m2 ) •
l
(f og)0(m1 ,m2),
and
(f(ml +n 1 ),f(m2+n 2»
Thus
fo«m1,m2)+(n1,n2»· fo(m1+n1,m2+n2) •
=
(fml+fn l , fm 2+fn 2)
Q is a left vector space over
Theorem 7.
F such that
If
Proof:
F and the dimension is clearly 2.
Q contains a subfie1d
Q is a left vector space of dimension 2 over
Let
the values of
x
s
~
coordinatized by
m , m
2
1
and identifying
see that the set of
Sm
F,
in
F,
then the dual
Q is derivable.
Restricting
m • {(x,xom): xeQ}.
we define sm· {«f,0),(f,0)o(m ,m »:feF}
l 2
• {(O,x): xeF},
to those of
m = (m ,m ),
1 2
f ,f 2 ,f eF}
1
3
= fo(m1,m2)+fo(n1,n2)·
Q is a right quasifie1d and if
of the translation plane T
where
~
F.
and
The
s
sm all lie in
sm· F(1,0,m ,m2)
1
IT
= {(f1 ,0,f 2 ,f 3):
with the pair
(m1 ,m )
2
form the set of points of the affine plane over
Thus the conditions of Theorem 5 are satisfied so that
aT
we
F.
is derivable.
Theorems 6 and 7 yield an algebraic characterization of derivable shears
planes first proved by Ostrom [6].
13
§4.
~LES OF DERIVABLE SHEARS PLANES
A semifie1d is a right quasifie1d satisfying the left distributive law.
Thus if
by
T is coordinatized by a semifie1d Q,
then
Q* which will also be a translation plane.
middle and right nucleus of
6T
may be coordinatized
In a semifie1d define the left.
Q as
Nt = {a: ao(boc)· (aob)oc for all b,ceQ}
Nm •
Nr
=
{b:
ao(boc)
= (aob)oc
for all
a,ceQ}
{c:
ao(boc)· (aob)oc
for all
a,beQ}.
Since the left distributive law is always valid, the kernel of
nucleus and the kernel of
Q*
Q is the left
is the right nucleus of Q.
A weak nucleus of Q is a field
F such that the equation
(aob)oc holds whenever at least 2 of
a,b,c
ao(boc)·
F.
are in
Knuth [4] has defined a class of semifie1ds of degree 2 over their weak
nucleus.
Since both
Q and
Q*
their weak nucleus the planes
tively are derivable.
are thus 2 dimensional vector spaces over
6T
and
T defined over Q*
and
Q respec-
investigation of the derived planes, which will be
An
translation planes [Dembowski, p.224] and thus not strict semitrans1ation planes,
will be undertaken at a later time.
In this present paper, we shall define one
subset of Knuth's planes and make some observations about the representation
of the derived planes.
Let
and let
m
F • GF(p )
a,a,a
p
an odd prime,
be automorphisms of
F
m > 1.
F
define:
(a,b) + (c,d)
•
(at-c, b+d)
(a,b)
•
(aC+b d f, a d+bc).
(c,d)
a
a
f
be a nonsquare in
not all of which are the identity.
the set of ordered pairs of elements of
0
Let
a
F
On
14
Then this defines a semi field
{(f,O):
feF}
~
F
Q
of dimension 2 over its weak nucleus
[Knuth, p.2l3].
We next determine the left and right nucleus of Q.
Suppose
(m,n)
in Q,
(g,h)
is in the right nucleus of Q.
we have
Then for all
(a,b),
«a,b)o(g,h»o(m,n). (a,b)o«g,h)o(m,n».
Thus
1)
«a,b)o(g,h»o(m,n). (ag+b~8f, aOh+bg)o(m,n) •
«ag+b~8f)m+(aoh+bg)an8f, (ag+b~af)°n+(aoh+bg)m)
and
2)
(a,b)o«g,h)o(m,n» • (a,b).[gm+ha n 6f, g0 n+hm] •
(a(gm+han 6f)+b a (g0n+hm) 8
f, 0
a 0
(g n+hm)+b(gm+ha n 8f».
Equating the first components of 1) and 2), we find
3)
b~Bmf + ao~an8f + ba gan 8f • ah an 8 f + b a g06n 6f + b~8m6f.
In 3), setting h. 0 we find ba gan 6f . ba g06n 6f.
4)
For this to hold for all b, g,
In 3) setting
5)
g • a • 0,
or a. 08.
we find ba h 8mf • b~am8f.
we must have m • m8 •
ao~an8f • aha n 8f
For this to hold for all b, h,
In 3) setting b • 0,
6)
we have either n· 0
we find
so that either n· 0
or else oa· 1.
Thus the first components are equal if ma • m and either n· 0 or else
a • oB
and oa· 1.
The converse is easily verified.
Equating the second components of 1) and 2), we obtain
7)
baohBonfo. bhanBf.
This equation will hold for all b, h if and only if either n· 0
nfo • nBf and ao. 1, 80· a.
Lemma 4.1.
and 0
= a-I
The right nucleus of Q is
{(m,O): m6.m}
and then the right nucleus of Q is
or
B. a 2
{(m,n): m8.m, and nfo.n6f}.
unless
15
Note that the second condition does not guarantee that there will be a
nonzero
n satisfying the given equation.
an example, let e be a primitive element of the field GF(p12). F
3
6
9
2j 1 for some j since
a: x+xP , a: x~xP, a: x+xP • W~ may write f . e +
a
f is a non-square in F. Write n = ei if n ~ O. Then nf - naf becomes
ei .e(2j+1)p9 • ei •p6 e2j +1 or i+p9(2j+1) = p6 i +2j +1 mod(p12_ 1 ) or
As
(p9_ l )(2j+1) _ (p6_ l )i mod(pl2_ 1 )
and only if p6_ l
divides
and this congruence has a solution
(p9_ 1 ) (2j+1)
which holds if and only if
i
p3+1
divides
(p6+p3+1 )(2j+1). But p3+1 is even and the other term is odd.
the only solution to nfa • naf is n· O.
A similar argument from equations 3) and 7) assuming
a
and b
if
So
to be
fixed yields
The left nucleus of Q is
LeTmla 4.2.
a - a
{(a,O): a aa-a}
unless
a = 1,
and then the left nucleus of Q is
{(a,b):
Examples:
Let
F
= GF(p 6 ):
4
a: x-+xP ,
2
a: x:+xP •
{(f,O): f€F}
and the right nucleus of
2
2
with N nF • {(f,O): fP af} ~ GF(p ).
Then the left nucleus of Q is
4
Q is isomorphic to GF(p)
2
a: x+xP ,
Thus the translation planes
r
T and
oT
coordinatized by Q and Q*
respectively can be represented in the usual manner in 4 space over
F and in
6 space over Nr •
There are 2 extremes in choosing the space over which to represent a
given translation plane.
One is to represent it over the kernel so that the
projective space is of lowest possible dimension.
over the prime field.
The other is to represent it
The advantages of the latter method occur in studying
col1ineations of the plane, for then all co11ineations are given by linear
16
transformations.
Another advantage is found in Theorem 2 and the enunciation
of Theorem 5.
In order to preserve the representation of Baer subp1anes as linear spaces,
and so preserve the representation of Theorem 5, we must represent the plane
over a subfie1d of the kernel intersect
F.
a semitrans1ation plane in 4 space since
represented in 12 space over
2
GF(p ).
Thus
60T
may be represented as
Nt.
But
6T· 600T
F·
Note that
kernel and there are no subspaces of 6 space over
that the subplane of
oT
coordinatized by
oT
is of degree 3 over its
GF(p4)
{(f,O): feF}
sented by a linear subspace in this representation of
must be
with
p6
cannot be
points so
repre~
oT.
The author is not aware of any strict semi translation planes derived from
shears planes of degree greater than 2 over their kernel.
Referenaes:
[1]
Bose, R.C. and Bar10tti, A.: "Linear representation of a class of projective planes in 4 dimensional projective space." Ann. Mat. Pura.
AppZ.~ 88, (1971) 9-32.
[2]
Bruck, R.H. and Bose, R.C.: "Linear representations of projective planes
in projective spaces." J. AZg. 4 (1969) 85-102.
[3]
Dembowski, P.:
1968.
[4]
Knuth, D.: "Finite semifie1ds and projective planes." J. AZg. 2 (1965)
182-217.
[5]
Ostrom, T.G.:
[6]
Finite
Geometries~
Springer Verlag,
New York/Berlin
"Semitrans1ation planes." Trans. A.M.S. 111 (1964) 1-18.
"Vector spaces and construction of finite projective planes."
Arch. Math. 19 (1968) 1-25.