I nternat. J. Math. & Math. Sci.
(1982) 159-164
Vol. 5 No.
159
ON ELATIONS IN SEMI-TRANSITIVE PLANES
N.L. JOHNSON
Department of Mathematics
The University of Iowa
52242
Iowa City, Iowa
U.S.A.
(Received January 2, 1980)
ABSTRACT.
be a semi-transitive translation plane of even order with
Let
reference to the subplane
axis in
0
0"
admits an affine elation fixing
If
and the order of n O is not 2 or 8,
KEY WORDS AND PHRASES.
then
for each
is a Hall plane.
Elons, Semi-transitive planes.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
0
50D05, 05B25.
INTRODUCTION.
Kirkpatrick [ 9] and Rahilly [ i0] have characterized the Hall planes as those
generalized Hall planes of order
In
q
2
that admit
q+l
central involutions.
7] the author has shown that the derived semifield planes of characteristic
and order
volutions.
q
2
are Hall planes precisely when they admit
q+l
central in-
This extends Kirkpatrick and Rahilly’s work as generalized Hall planes
are certain derived semifield planes.
If a translation plane
distinctaxes
of order
then the generated group
a normal subgroup
N
little is known about
q
2
admits
contains
q+l
affine elations with
SL(2,q)
of odd order and index 2 (Hering [5]).
except that it is usually dihedral.
Sz(q)
or contains
In the latter case,
N.L. JOHNSON
160
In this article, we study semi-transltlve translation planes of order
q+l
that admit
q
2
affine elations.
In [8], the author introduces the concept of the generalized Hall planes of
type
These are derivable translation planes that admit a particular colllnea-
i.
tion group which is transitive on the components outside the derivable net.
In
this situation the group is generated by Baer colllneatlons.
More generally, Jha [6] has considered the "seml-transltlve" translation
planes.
(i.i)
Let
be a translation plane with subplane
such that
llneation group
i)
fixes
pointwise,
0
2)
leaves
3)
acts transitively on
then
If there is a col-
0"
invarlant, and
0
-0
is said to be a semi-transltlve translation
plane with reference t__o
0
and with respect to
Our main result is that seml-transltlve planes of order not 16 or 64 that
admit elations with axis
planes.
2
q #64
admitting
of
0
are Hall
(2.1) THEOREM.
q+l
Let
elations with distinct axes is derivable.
q+l
TRANSLATION PLANES OF EVEN ORDER
admits
for every component
0
We also give a necessary and sufficient condition that a translation
plane of order
2.
fixing
q
2
ADMITTING
q+l ELATIONS.
be a translation plane of even order
Let
afflne elations with distinct axes.
SL(2,q)
either isomorphic to
Then
invariant.
2
# 64
that
denote the net of degree
q+l that is defined by the elation axes and assume the group
these elations leaves
q
D
generated by
D
is derivable if and only if
or is dihedral of order
2(q+l)
is
where the cyclic
stem fixes at least two components.
PROOF.
If
D
is Desarguesian by
Let
Y
D
<,X
SL(2,q)
is isomorphic to
-
(xT,yT
-I)
is derivable and actually
Foulser-Johnson-Ostrom [3].
s2=xq+I=I,x=x -is >.
for some matrix
T
<X>
If
O then we may choose coordinates so that
(x,y)
then
of order
s
is
q+l.
(x ,y) --+ (y ,x)
and
O,
X
fixes the components
X
is
161
ELATIONS IN SEMI-TRANSITIVE PLANES
.
E
By Ostrom [ii], Theorem 3, there is a Desarguesian plane
x-fixed components and
two
.
able in
fix each Baer subplane of
D < GL(2,q)
By Gleason [4],
SL(2,q)).
D
Thus,
is derivable then
that
II
Then
X
SL(2,q)
(each elation
.
.-
q 2_1
containing the
-
permutes the components of
E A
teristic in
<X>),
is a collineation group of
E.
(xa,ya)
E
(Note X fixes
?]
componentwise.)
Since
ordinates so that the components of
then
fixes
X
<X2>
<X>,
x=
we may assume
i.
X
O
are
e GF(q
for all
is
Thus,
2
odd,
Y
GF(q
least two
cponents of
2)
and
(X 2 >
<X >.
O, y
xa,
=
=
E
of
fixes
.
X),
X
has the
GF(q2).
a
Choosing coGF
.
if d only if
X
# 64).
is charac-
The collineation
share at least two components (those fixed by
and
3.
y
q+l
2
is greater
<X>
(i.e.,
is an automorphism of
where
q
such
(see Ostrom [ii], CoT.
of
x-fixed components
Since
(x,y)--+
replace
so there is a unique Desar-
q+l).
form
DI"
Let
(one exists since
to Theorem 1--uniqueness comes from the fact that the degree of
than
is then in
<X> <X>
Let
componentwise in the derived plane
Y
must
Moreover, if
2(q+l).
or is dihedral of order
fixes at least two infinite points of
D
q+l
is transitive on the elation axes so
is a prime 2-primitive divisor of
guesian plane
,
By Foulser [2], Theorem 3,
fixes at least two infinite points of
X
fixes
so
D
SL(2,q)
is clearly
O.
D
so that
and thus deriv-
Since each elation fixes
incident with
in its action on
E
is an Andr net in
Clearly
is derivable.
Conversely, suppose
containing the
(q2)
Since
pointwise.
Since
must fix at
X
SEMI-TRANSITIVE TRANSLATION PLANES OF EVEN ORDER.
Let
be a translation plane of even order
as in section 2.
Then,
q
2
that admits
denote the net defined by the elation axes.
that commutes with
(3.1) THEOREM.
admits
q+l
D.
Let
elations
is a derivable plane provided the generated group
SL(2,q).
is dihedral and the cyclic stem fixes at least 2 points or
let
q+l
Then clearly,
Let
In any case
,$ be a collection group
must fix
pointwise.
be a translation plane of even order
elations with distinct axes.
D
Assume the group
D
q
2
# 64
that
generated by these
162
N.L. JOHNSON
of the elation axes Invariant.
elations leaves the net
q+l
D
llneatlon group which commutes with
and is transitive on
is a Hall plane.
PROOF.
-
m.
2
@ 64,
IDI.
q+l
Clearly,
q(q-l)
acts on the
X
,
N
points of
X
-
-
there is a prime 2-prlmltive divisor
--
By Gleason [4],
order
q
Since
Since
points of
commutes with
must fix
X
pointwise.
be a col-
N
Then
N
q2_l
of
m
D
be an element of
Let
q+l.
m
,
Let
of
so must fix at least two
is transitive on
and
Z
By the corollary to Theorem i, Ostrom [ii], there is a Desargueslan plane
and
.
E--D
U?
Let
where
is the net complementary to
of degree q+l.
for some net
in
.
Then
are two extensions of a net
So I and
of critical deficiency (see Ostrom [12]).
and
must be Hall since
Then
cannot be itself Desarguesian)by Ostrom
must be related by derivation (i.e.,
Z
are exactly the common components of
such that the components fixed by X in
[12].
The conditions of (3.1) are close to giving the definition of a "seml-transl-
tive" translation plane (see (i.i)).
satisfy condition 2.
In (3.1), it is possible that
may not
Also, it is not clear that a seml-transitive translation
However, Jha [6] shows if
plane is derivable.
,
nontrlvial kern homology in
then
has order not 16 and there is a
is derivable and
0
is a Beer subplane.
We may overcome this restriction on the kern in our situation:
(3.2) THEOREM.
Let
with respect to a collineation group
Let
and with reference to a subplane
admit an affine elation for each axis in
i)
If the order of
0
is not 8 then
2)
If the order of
0
is not 2 or 8
PROOF.
collineation group of
must leave
(Note
pointwise.
Clearly,
0
n
0
i
ISnl
is divisible by
is a Beer subplane of
i
is a Hall plane.
be a minimal subplane of
i
with reference to
pointwise fixed since
0.
then
Clearly, the stabilizer
0
n
0"
is derivable.
Following Jha’s [6] ideas, let
erly containing
i
.
be a semi-transitive translation plane of even order
i
0"
of
i
prop-
is a semi-transitive
Moreover, a sylow 2-subgroup of
fixes
0
(2r+l)-(2S+l)
and fixes
for some
n
0
N
r,s.)
163
ELATIONS IN SEMI-TRANSITIVE PLANES
Every elation which leaves
So the group
variant.
,
clearly,
D
invariant must also leave any superplane in-
0
generated by the elations leaves
commutes with
D
,
since
fixes
mute with each central collineation fixing
By (3.1),
if the order of
0
0
N
i
invariant and,
pointwlse
(, must com-
0
is not 8 then
i
is a Hall plane and
derivable.
We may now directly use Jha [6] to show that if the order of
no___t 2’ then
i
is
i
0
is
(that is, Jha uses the hypothesis that there is a kern homology
is derivable).
i
Actually, our proof of (3.2) proves the following more general theorem for
to show that
arbitrary order.
(3.3) THEOREM.
to
0
Let
and with respect to
be a semi-transitive translation plane with reference
r
Let X be a collineation generated by
and order p
central collineations leaving
divisor of (order
0
0
2_ 1 (where
invariant such that
the order of
0
IXI
is not 2).
Is a prime p-primitive
Then n is a Hall plane.
Note that a semi-transitive plane of odd order p 2r must admit Baer p-elements
(see Jha [6]).
By Foulser [i], we could then not have both Baer p-elements and
elations so we could restate our Theorem (3.2) without reference to order.
(3.2)2) is also valid if the order
0
is 8.
The arguments supporting this will
appear in a related article.
REFERENCES
i.
Foulser, D. A.
354-366.
2.
Foulser, D. A.
3.
Foulser, D. A., N. L. Johnson and T. G. Ostrom.
Baer p-elements in translation planes, J. Algebra 31(1974)
Subplanes of partial spreads in translation planes, Bull.
London Math. Soc. 4(1972) 1-7.
Desarguesian and Hall planes of order
4.
Gleason, A. M.
5.
Hering, Ch.
27(1972)
q2
Characterization of the
by SL(q,2), submitted.
Finite Fano planes, Amer. J. Math.
78(1956)
797-806.
On shears of translation planes, Abh. Math. Sem. Hamburg
258-268.
6.
Jha, V.
7.
Johnson, N. L.
Finite semi-transitive translation planes, to appear.
11/2(1978)
On central collineations of derived semifield planes, J. Geom.
139-149.
164
8.
N.L. JOHNSON
Johnson, N. L.
Distortion and generalized Hall planes, Geom. Ded.
4(1975)
1-20.
9.
Kirkpatrick, P. B. A characterization of the Hall planes of odd order, Bull.
Austral. Math. Soc. 6(1972) 407-415.
I0.
Rahilly, A. Finite collineation groups and their collineation groups, Thesis,
Univ. of Sydney, 1973.
ii.
Ostrom, T. G.
J.
12.
Alsebra
Ostrom, T. G.
1381-1387.
Linear transformations and collineations of translation planes,
(3) 14(1970)
405-416.
Nets with critical deficiency, Pacific J. Math. 14(1964),