Internat. J. Math. & Math. Sci.
Vol. 6 No. 3 (1983) 605-608
605
RESEARCH NOTES
A CHARACTERIZATION OF THE DESARGUESIAN PLANES
OF ORDER q2 BY SL(2,q)
D.A. FOULSER
Mathematics Department
University of lllinois at Chicago Circle
Chicago, lllinois 60680
Department of Mathematics
The University of lowa
lowa City, Iowa 52242
T.G. OSTROM
Department of Pure and Applied Mathematics
Washington State University
Pullman, Washington
99164
(Received January 25, 1982)
ABSTRACT.
The main result is that if the translation complement of a translation
plane of order
q
order
are
q
2
contains a group isomorphic to
elations (shears),
SL(2,q)
and if the subgroups of
then the plane is Desarguesian.
This generalizes
GF(q).
earlier work of Walker, who assumed that the kernel of the plane contained
KEY WORDS AND PHRASES.
Translation planes, tanslion compleme, elons.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
THEOREM.
prime.
Let G
51A40, 20B25.
2
Let w be a translation plane of order q
SL(2,q)
where q
p
r
and p is a
be a subgroup of the translation complement of w whose ele-
ments of order p are elations.
Then w is a Desarguesian plane.
-
This theorem is a special case required in the classification of all translation
planes w of order q
2
which admit a collineation group G
SL(2,q) [i, 2].
sification is a generalization of the work of Walker and Schaeffer
[3, 4],
That claswho
assume, in addition, that the kernel of w contains GF(q).
To begin the proof, let W be a vector space of dimension 2r over GF(p).
Since
606
D.A. FOULSER, N. L. JOHNSON and T. G. OSTROM
is a
4r-dimensional vector space over GF(p), we may represent
lines containing
GF(p)--linear
(x,y), where x,y E W.
are vectors
the points of
(0,0))
have the form
transformations A: W
fining equations x
[(0,y)"
W}
xA, respectively.
0 and y
(i.e.,
The components of
(x,xA)" x E W}
and
the
for various
We will denote the components by their de-
W.
/
y E
as W @ W so that
Next, note that each Sylow p-sub-
group Q of G is abelian and hence all the elements (# l) of Q have the same elation
ISl
axes; thus
q +l and
where K is a field of 2r
tion axes
(that is,
LEMMA 2.
isfied:
INI
q+l.
(Hering [5], Ostrom [6]).
LEMMA i.
PROOF.
slq
N)
-
as above such that
GF(q).
Further, the ela-
xA (A E K) and x
0.
G such that the following conditions are sat-
Ipt-1
(ii)Igl
and K
have the form y
There is an element g
2
for t
< Pr;
and
(iii)g fixes
a component
pt -I
-i, and s
8 or q
Igl
g such that
S\N.
p and p +I
Igl
9.
Finally, if q
Since
IS\NI
LEMMA 3.
2
p and p +l
SL(2,K),
has orbits of lengths i, 2, and
length i or 2 on
56
S\N.
Choose g
and
L(g)
h
2
g
If q
s sc
(iii)
8,
choose
4 p(p-l)
fixes an element of
7.
h
2
Then
Hence, h
then h has an orbit of
S\N.
G so that g satisfies the conditions of Lemma 2, and let
GF(p).
Then
L(g)
is a field
contains the subfield
K:=
PROOF.
S\N.
fixes every component of
be the ring of matrices generated by g over
GF(q
in
gl
(mod 3), g must fix one of the elements of
in S, and since
Therefore g
In the first case, let
choose h of order 8 in G and let g
so g
4
[7].
q -I has a p-primitive prime
q(q-l) components
0
a
a
Then g also satisfies condition
is a prime and g permutes the
has order 2 in G
L(g)
2
(i) and (ii).
2
slq +i).
2r (hence
t
for 0
that g satisfies conditions
because
-1 if s is a prime,
The integer s is a p-primitive prime divisor of q
divisor s unless q
-
GF(p)
2r matrices over
the elements of
(i)Igllq+l
We may coordinatize
which is not in the set N.
of
g
and let N be the subset of elatior
Let S denote the set of all components of
axis.
G c
GL(2,K)
by Lemma i.
A(
As a 2
2 matrix over
K, g has a minimum
CHARACTERIZATION OF THE DESARGUESIAN PLANES
polynomial
f(x) over K
of degree g 2.
and f is irreducible over K.
contains
L(g)
Since
607
[
Igl
q(q-l),
Therefore, g and K generate a field U
as a subfield.
Since
IPt-
Igl
(for t < 2r),
i
-
then the degree of f is 2
GF(q’2
L(g)
then
which
U and
L(g)
LEMMA 4.
field isomorphic to
V(2r,p)
space V
I[0 A0]. K1
A
hence
A E
of dimension 2r over
of T over K has degree < 2.
components y
GF(p).
Hence,
affine plane of order
x,y E
GL(2,L) acts
the points of
L}
and
and the components of
.
are components both of
*
and
.
w*
and of
q(q-l) components
of
*
*
SL(2,K)
y
*
*.
of a component of
distinct
Therefore, T
.
Since G=
are
[y
xC- C E
*
i.e., the points of
L} U
SL(2,K)
GL(2,L),
(for
Ix
0}.
Clearly,
on w by identifying
L, the components y
Finally, recall that
outside of N
denote the Desarguesian
KIT];
We superimpose
Since K c L and T
collineation group of
the
w*
Let
coordinatized by the field L
as a collineation group of
w*
K, then the
GF(q ).
We can now complete the proof of the Theorem.
[(x,y)"
the minimum polynomial
If T has an eigenvalue A in
-
KIT]
thus K
K makes V into a 2-dimensional vector
xA of w must intersect, which is impossible.
xT and y
is irreducible over K and
are
is a
2r matrices which act on a vector
space and T acts as a K-linear transformation of V.
f(x)
KIT]
xT,
fix the component y
T and the elements of K are 2r
centralizes T.
Then
GF(q ).
L(g)and
PROOF.
xT of S\N.
Let g of Lemma 2 fix the component y
xA of N and y
xT
then G acts both as a
SL(2,K)
acts transitively on
example, the stabilizer subgroup in
outside N has order q +i).
xT under G constitute q(q-l) components both of
*
Therefore, the images of
and of
; so
as
required.
The research of all three authors was partially supported by the National
Science Foundation.
REFERENCES
i.
FOULSER, D.A. and JOHNSON, N.L.
SL(2,q)
2.
2
"The Translation Planes of Order q that Admit
as a Collineation Group, l, Even
Order," to appear
in J. of Algebra.
2
"The Translation Planes of Order q that Admit
SL(2,q) as a Collineation Group, II, Odd Order" to appear in J. of Geometry.
FOULSER, D.A. and JOHNSON, N.L.
D.A. FOULSER, N. L. JOHNSON and To G. OSTROM
608
"A Characterization
49, 216-233, 1979.
3.
WALKER, M.
4.
SCHAEFFER, H.
Thesis
5.
HERING, C.
1972.
6.
0STROM, T.G.
of Some Translation
Planes" Abh. Math. Sem. Hamb.
"Translations Ebenen auf denen die Gruppe SL(2,p r) operiert"
(Tubingen, 1975).
"On
Shears of Translation Planes" Abh. Math. Sem. Hamb.
3_7, 258-268,
"Linear Transformations and Collineations of Translation Planes"
J._ Algebra 14, 405-416- 1970.
7.
ZSIGMONDY, K.
1892.
"Sur
Theorie der
Potenzreste"
Monatsch. Math. Phys. 3,
265-284,