Internat. J. Math. & Math. Sci.
Vol 9. No. 3 (1986) 617-620
617
RESEARCH NOTES
ON COLLINEATION GROUPS OF
TRANSLATION PLANES OF ORDER q4
V. JHA and N.L. JOHNSON
Department of Mathematics
The University of Iowa
Iowa City, IA 52212 U.S.A.
(Received April 9, 1984 and in revised form March 31, 1986)
ABSTRACT.
Let
ble group
G
the structure of
and
L2(2 s)
G
q
be an affine translation plane of order
P
in its translation complement.
G
G
s
admitting a nonsolva-
fixes more than
G
In particular, if
is determined.
for some integer
If
q+l
is simple then
slopes,
q
is even
at least 2.
KEY WORDS AND PHRASES. Translation planes, nonsolvable groups.
19B0 MATHEMATICS SUBJECT CLASSIFICATION CODES. 50D05, 05B25
1.
INTRODUCTION.
Let W
be a translation plane of square order
SL(2,pt)
tion group isomorphic to
p
2r
If
admits a collinea-
N
and the Sylow p-subgroups are planar, then usually
(in the known cases) the group fixes
pr+l
components (or slopes).
Generally, how-
ever, simply knowing that a group fixes a number of slopes says essentially nothing
we can
concerning the structure of the group. However, for planes of order q
That is, in this note our objective is to prove the following.
admitting a nonTHEOREM A. Let W be an affine translation plane of order q
than q+l
G
more
fixes
Suppose
complement.
solvable group G in its translation
make some progress.
slopes.
(1)
If
q
81 G
is odd then
and the
of
G
G
Furthermore,
2.
al-
ways contains the kern involution.
(2)
If
q
for some
s,
and
elementwise.
so
N
G/N
fixes exactly
s
N
contains a normal subgroup
is of odd order.
N
such that
L2(2s),
Now the Sylow 2-subgroups fix Baer subplanes
If
q2+l
G
slopes.
is simple then
q
is even and
G
L2(2 s)
for some inte-
2.
PROOF.
simple.
G
Furthermore, the Baer subplanes share the same points at infinity and
COROLLARY 1.
ger
is even then
When
q
is odd the kern involution is central in
G
and so
G
is not
618
V. JHA and N. L. JOHNSON
(1)
REMARKS.
q
When
G’s
is odd it is possible to find
with 2-rank one and
2-rank two such that they satisfy the hypothesis of Theorem A.
G
SL(2,q),
1
(Gl,>
(2)
with 2-rank two.
Both the theorem and its corollary cease to be unconditionally true if we
"at
to fix
G
allow
’G’
is any Baer involution, we get a
where
For example,
has 2-rank one; if we now choose
q
acting on Hall planes of order
least
slopes", instead of "more than
q+l
only known counterexamples seem to be the Lorimer-Rahilly planes
q+l
Ill
slopes":
the
and its trans-
pose, the Johnson-Walker plane [2].
[3]
The following well known consequences of Foulser
will be used on several
occasions in the proof of Theorem A.
RESULT 0.
p
order
Suppose
B
is a collineation group of an affine translation plane of
that fixes a Baer subplane elementwise.
(i)
(ii)
(iii)
B
B
the Sylow p-subgroups of
the Hall
p
are elementary abelian; and
B
subgroups of
2.
PROOF OF THEOREM A.
and
8orresponds to Ostrom’s ideas in [].
are cyclic.
We begin by dealing with the case when
LEMMA 1.
q
If
;
kern involution of
PROOF.
Let
K.
that
Then
is solvable;
is odd then any Klein
,
4-group
]
For any involution
x
G
in
must contain the unique
G
be a Klein group in
in
K rite W
x
.
The first step is folklore
is odd.
we shall always denote this involution by
[1, a,
K
q
and suppose, if possible
for its fixed Baer subplane.
Now we claim w N
cannot be a subplane of
if x,y are distinct involutions
x
y
in K. If W
we have a Klein group fixing elementwise a Baer subplane of W,
W
y
x
contrary to Result O. So W N
is a fourth root subplane of W and now we contray
x
dict the assumption that G fixes more than q+l slopes. Thus xy acts like -1
G,
on the fixed components of
in the spread associated with
;
i.e.,
xy
is the
required involution.
LEMMA 2.
PROOF.
q
If
IGI
G
is odd then
[5,
For the same reason the Sylow 2-subgroups of
IGI
G
G
when
LEMMA 3.
G/(: )
and
8
Suppose
I GI"
q
is solvable.
.
cannot be cyclic of order
only if the Sylow 2-subgroups are Klein groups.
the kern involution
bility of
8.
then, by Burnside’s theorem, G has a normal
6.2.11] and we contradict the assumption that G is nonsolvable.
If 2 exactly divides
2-complement
is divisible by
So by Lemma l,
G
Hence
contains
Thus we contradict the nonsolva-
The result follows.
is odd.
Then
G
cannot contain an elementary abelian
2-group of order 8.
PROOF.
If
S
is an elementary abelian subgroup of
G,
whose order is
8,
we
may write
s
{,,,,,y,, ]
(.)
COLLINEATION GROUPS OF TRANSLATION PLANES
619
and assume that
(2.2)
.
,
4.
But by Lemma i, both
L
and
Interchanging the role of
@
and
are distinct subgroups of order
,
M
contain the kern
involution
:.
Hence
is also
:.
The lemma follows, since we have contradicted the assumption that
LEMMA 4.
G
contains
Let
PROOF.
S
G
cyclic, because
one.
So
2)
the quaternion group of order
8.
denote the unique involution in
Q
U
a
Q
because
U
for
non-
S
is
(2.3)
and, to get a contradiction,
Now
>
fixes
Q
u
leaves
invariant
Moreover, no element of
but does not fix it elementwise because of Result 0.
induce a Baer involution on
8 (Lemma 2) and
n m
(x2n-2 y),
is a Baer involution with fixed plane
Now let
S
Now Lemma i applies unless the 2-rank of
is nonsolvable.
Q
and so contains
assllme
G.
is the generalized quaternion group
n
n-I
2
2
-i
-i
y xy
(x,y x 2
x
x
i, y
S
Thus
u.
the kern involution of
denote a Sylow 2-subgroup of
we find that
q+l
u
slopes of
Q
can
Thus the
restriction map
> Q
Q
<),
has as its image
(2.4)
Z of
order
4, contrary to
q
(i).
is
[6],
deduced
[7].
RESULT 5.
Suppose
H
nonsolvable group
is an affine translation plane of even order admitting a
@
in its translation complement.
fixes a Baer subplane elementwise.
6.
contains a normal subgroup
[6,
Theorem
N
H
such that
s.
2.3].
is even then the Sylow 2-subgroups of
q
If
Assume a Sylow 2-subgroup of
for some integer
Use Johnson’s argument in
PROOF.
H
Then
L2(2 s)
N
is of odd order and
L,94A
p
To deal with the case
is even we need the following version of a theorem of Johnson
from Hering
H/N
ker
Result 0.
The lemmas proved so far add up to Theorem A, Part
when
So
is the kern involution of
where
clearly a noncyclic group
I
G
fix Baer subplanes of
elementwise.
U
Let
PROOF.
by
S
be a Sylow 2-subgroup of
form a subplane
assume
U
S
u
UDUS"
S’
because
S
is not a Baer subplane of
S
and let
of
D
S
.
fixes many slopes.
Hence the order of
q+l
slopes.
S&
q
U
fixed
To get a contradiction we
be any involution in the center
Now let
Then we have a chain of planes
be its fixed Baer subplane.
fixes more than
and note that elements of
G
and we contradict the assumption that
G
The result follows.
Now by Result 5 and Lemma 6 we immediately have
LEMMA 7-
Suppose
q
is even.
G
Then the Sylow 2-subg:’oups of
planes elementwise and generate of a subgroup
N
L2(2s)
where
2
s
fix Zaer sub-
,I IGI
V. JHA and N. L. JOHNSON
620
To complete the proof of Theorem A we restrict ourselves from now on to the situation described in Lemma
LEMMA 8.
PROOF.
N
7.
fixes a unique affine point
Suppose
.
N,
cond affine point of
0.
which is in the translation complement of
Then
N
is a planar group of
W
of order
N,
>
fixes a se-
q.
Now by
D
Lemma 7 we have a Baer chain
we have the same contradiction
If
#
as in Lemma 6; otherwise we have a nonsolvable group fixing a Baer subplane element-
S NN"
N NS
wise, contrary to Result 0.
LEMMA 9.
The only affine point fixed by distinct Sylow 2-subgroups of
the unique point fixed by
PROOF.
N
0,
is
N.
is generated by any two of its Sylow 2-subgroups, so we contradict
N
Lemma 8 unless the lemma is valid.
2s#
For
shows than
s= 4
4, Lemma 9,
N
if
2
i.
LORIMER, P.
as
q2+l
when combined with Foulser and Johnson
[8,
Proposition
3.4],
From Johnson [9, Theorem 2.1], the same is true
fixes at least 3 slopes. This proves Part (2) of Theorem A.
fixes
N
slopes.
A Projective Plane of Order 16, J. Combinatorial Theory A16 (1974),
334-347.
2.
WALKF, M.
Soc. 8
A Note on Tangentially Transitive Affine Planes, Bull. London Math.
(1976), 273-277.
3.
FOULSER, D.A.
4.
OSTROM, T.G.
Subplanes of Partial Spreads in Translation Planes, Bull. London
Math. Soc.
4 (1972), 32-38.
Elementary Abelian 2-Groups in Finite Translation Planes, Arch. Mat.
36 (1981).
5.
6.
SCOTT, W.R. Group Theory, Prentice Hall, New Jersey, 1964.
JOHNSON, N.L. The Geometry of GL(2,q) in Translation Planes of Even Order
7.
Internat. J. Math. & Math. Sci. i (1978), 447-458.
HERING, Ch. On Subgroups with Trivial Normalizer Intersection, J. Algebra 2__0
q
(1972), 622-629.
8.
9.
FOULSER, D.A. and JOHNSON, N.L. The Translation Planes of Order p2 that Admit
SL(2,q) as a Collineation Group. II. Odd Order, J. Geometry 18 (1982),
122-139.
JOHNSON, N.L. Representation of SL(2,4)
J. Geometry 21 (1983), 184-200.
on Translation Planes of Even Order,