Arch. Math., Vol. 63, 17-22 (1994)
0003-889X/94/6301-0017 $ 2.70/0 9 1994 Birkh/iuser Verlag, Basel
Semifield planes with a transitive autotopism group By MINERVA CORDERO *) and RAOL F. FIGUEROA t)
1. Introduction. Let n be a non-Desarguesian semifield plane of order p", where p > 3 is a prime number and n > 3. Let G denote the autotopism group of n relative to an autotopism triangle A. Suppose that G induces a group G transitive on the non-vertex points of a side of A. Then by [4] Lemma 4.2, we have that either (i) G < FL(I,p"), or (ii) p" = 34 (and G is a certain group), or (iii) p" = 36 and G ~ SL(2, 13). In this article we will show (see Theorem 4.2) that if either (i) or (ii) occurs, then n is a generalized twisted field plane. From this result it follows that the non-Desarguesian semifield planes of rank 3 are, in general, generalized twisted field planes. Another consequence is that it strengthens Theorem A in [5] and so we have that if a finite translation plane of odd characteristic admits a collineation group that fixes a point L on the line at infinity loo and acts 2-transitively on l~ - {L}, then the plane is Desarguesian or a generalized twisted field plane. In the next section we introduce some notation and background. In Section 3 we discuss the case when p" = 34. In Section 4 we prove (Theorem 4.1) that if G has an p"- 1 element of order - - , where #[ n, then n is a generalized twisted field plane. Indeed, /2 if case (i) G < FL(1, p") occurs, then G has an element as above. 2. Preliminaries. Let V denote the n-dimensional vector space over the Galois field GF(p) consisting of all the n-tuptes of the form (x(~ where x e K = GF(p"), x ( ~ x p' for 0 - < i N n - 1 , and p is a prime number. The group Aut(V) of all the automorphisms of V over GF(p) consists of the non-singular matrices of the form
(a0 M=
a.) n-1 .
ai"-
al
a(~
""
a(z"-1)
i
"
...
-
an i
an-2
" ' "
*) Research supported in part by NSF Grant No. DMS-9107372 t) Research supported in part by NSF Grants RII-9014056, component IV of the EPSCOR of Puerto Rico grant and ARO grant for Cornell MSI Archiv der Mathematik 63
2
18
M. CORDERO and R, E FIGUEROA
ARCH. MATH.
where a o , a t , . ~ ~ we d e n o t e this m a t r i x M by [a o,a~ . . . . . a,_~)~. W=VGV, V(oo)={(O, vl[vev} and V ( M ) - { ( v , oM)lv~V} for M ~ A u t ( V ) M = 0 (the zero matrix). In [2] we p r o v e d the following.
Let or
Result 1. L e t n be a non-Desarguesian semifield plane of order p". f t n admits a collineation go of order a p-primitive prime divisor h o f p" 1. i. e., h ~p" I and h ~t~ pi _ 1 for 1 < i <- n 1. then (i)
(ii)
There exists a spread set {V(0), V ( ~ ) } w { V ( M ( y ) ) ] y ~ K ~ ]br a in W such that V [ ~ ) is a shear axis and for y ~ K • = K - [0}, M ( y ) = [aoy~e~ ~. .., a, ~ y~..... ~]' z A u t ( V L The coefficients a o = 1, al, . . . . a,_ ~ belong to K and the integers eo = O. e 1 . . . . . e,,_ ~ are such that 0 < e~ <- n - 1. e~ +- i if a~ 4= O, and ei :r e i when i 4: j and a~ #: 0 = aj. go=(
C
OD). where C = [ 7 , 0 . . . . . O]' and D = [ ~ , O . . . . ,O]' for some y, 6 ~ K ,
both o f order h and 7 = 6 Suppose n o w that n o is a semifield plane that a d m i t s a spread set as the one in p a r t (i) of the result above. T h e n (K, + , o) is a pre-semifield associated with ~o, where 4- is t~--].
the a d d i t i o n on the field K a n d x o y = Z a~xt~JY ~'~, for x. y ~ K . N o t i c e t h a t ff there ~s ~=o exactly one a, = 0, for some r 4: 0, then from [1], (K, ~ . o) is a generalized twisted field. Let (0) and ( ~ ) d e n o t e the points on the line at infinity of n o associated with V(O) a n d Vtool, respectively. Let 9 -
where A = [~, 0 , . . . , O] ~ and B = [fi, 0 , . . . ,
] , for
some e,/3 ~ K. Then ,q m a p s the subspace V ( M I with M = [xo, x 1 . . . . , x,_ ~]~ e A u t W L onto the subspace V (N h where N = A - ~M B = f ~ x o, ~ fi x ~. . . . . c~(U~ii B x , _ ] ' . Hence g is an a u t o t o p i s m of 7co (fixing (0), (ca), a n d (0, 0It if and only if (/~Y~'
fl
for each i such that a, + 0.
3. The ease p " = 34. Let 7r be a n o n - D e s a r g u e s i a n semifield plane of o r d e r 34. Let G d e n o t e the a u t o t o p i s m g r o u p of n relative to an a u t o t o p i s m triangle A, A s s u m e that G induces a transitive g r o u p on the non-vertex points of a side of A. T h e n n a d m i t s an a u t o t o p i s m of order 5, a 3-primitive divisor of 34 - 1, and, as we now show, ~ is a generalized twisted field plane, Theorem 3.1. L e t n be a non-Desarguesian semifield plane of orderr 3 4 with an autotopism g o f order 5, a 3-primitive divisor o f 34 - 1. Then n is a generalized twisted f i e l d plane. P r o o f . By Result 1 we m a y assume that 9 = (\ A0 and B = [fl, 0, . . . . 0] t for some ~ , f i ~ G F ( 3 4 ) ,
OB) ' where A = [~, O, . . , 0 ] ~
[~[ = [fi] = 5
and e#fl,
Since ~ = f l r
Vol. 63, 1994
Semifield planes with a transitive autotopism group
19
for some integer r, 2 < r < 4, from (2.1) we have that /~(1-r)vo, = f l l - ~ p i ( p = 3), a n d hence pe, _ I ==- r ( p e' -- p l ) rood 5, for each i such that a i + 0. A direct c o m p u t a t i o n shows that if r = 2, t h e n (i, ei) = (2, 3) or (3, 1); if r = 3, t h e n (i, ei) = (1, 2) or (2, 1), a n d if r = 4, then (i, el) = (1, 3) or (3, 2). S u p p o s e that r = 3. T h e n the p r o d u c t o of the pre-semifield associated with = is x o y = x y + a l x ( X ) y (2) + azx(Z)y (t). T h u s if al = 0 or a z = 0, then ~z is a generalized twisted field plane. Suppose that a l =t= 0 4= a2. Since x o y 4 : 0 for all x, y e G F ( 3 *) • by replacing x by ty, then dividing by ty 12 a n d s u b s t i t u t i n g y by I we have that Z
(3.1)
z 1~ + a l t 2 -I- a c t s + O,
for all z, t E G F ( 3 4 ) • . I f a ~ = s 2 is a square, then by replacing t by s - i t we m a y assume that a 1 = 1. If al is a n o n - s q u a r e , we m a y assume that a l = 0, where 0 e G F ( 3 4 ) is a primitive element such that 0 4 + 0 2 + 2 = 0. We c a n express a 2 as a 2 -----70i, where 7 ~ G F ( 9 ) • a n d 0 -< i -< 9, a n d after replacing t by k t in (3.1), where k ~ G F ( 9 ) • we may assume t h a t T = l orT=0 l~ 2=0 iora 2=01~ I n the following tables, for each value of i we give a value for j such that a l t z + a2 t8 ~ G F ( 9 ) • where t = 0 ) a n d a 2 = 0 i or a z = 0 1 ~ Since for z e GF(34), z ~~ takes o n all the values in G F ( 9 ) , it follows t h e n that there do n o t exist two n o n - z e r o coefficients a~, az for which x o y + 0, for all x, y E G F ( 3 4 ) • , a n d therefore 7c is a generalized twisted field plane. Case a 1 = I
Case
a I =
i
0
1
2
3
4
5
6
7
8
9
a2=0 i
j
0
24
4
72
23
6
12
12
11
56
a2 = 0 1 ~
j
0
9
7
16
8
18
It
1
36
l
i
0
1
2
3
4
5
6
7
8
9
a2 = 0 i
j
2
7
28
2
6
5
0
5
2
0
a2 =
j
l
11
13
9
0
9
7
1
15
1
0
010+i
The case w h e n r = 2 (or r = 4) can be reduced to the previous case by replacing x by x 9 (x 3, resp.) a n d y b y y3 (y9, resp.). 4. M a i n result. Let n be a n o n - D e s a r g u e s i a n semifield p l a n e of order p", where p is a n odd prime n u m b e r a n d n > 3. Let G be the a u t o t o p i s m g r o u p relative to a n a u t o t o p i s m triangle A a n d let G be the g r o u p i n d u c e d by G o n a side of A. First we prove that if d has a n element of order - - , where # is a n integer dividing n, t h e n ~z is a # generalized twisted field plane. T h e n we use this to prove the m a i n t h e o r e m in this paper. 2*
20
M. COROEROand R. E FIGUEROA
ARCH.MA'Ftt.
Theorem 4.1. Let rc be a non-Desarguesian semifield plane of order p", where p is an odd prime number and n ~ 3. Let A be an autotopism triangle of ~ and let G be the group induced by the aueotopism group G of ~ on a side I of A. I f G has an eiement of p" - t order ,-, where # is an integer dividing n, then ~ is a generalized twisted field plane. # P r o o f, Let 9 be an element in G such t h a t ~ ~ G has tile given order. Let h be a p-primitive prime divisor of p" - t (such an h exists by [6]). Since h X n we have that /p" - ] h X ~ and therefore ht--T[-~ ..... IT]- Then there is g o ~ ( g ) of order h. Thus w e c a n j
r-
apply Result J. So we assume that (0), (oo), and O = (0, 0) are the vertices of A, that rE admits a spread set as the one in part (i) of Result i, and that the matrix representa., tion of go has the form given in part (ii) of Result 1. Since g fixes the vertices of A w e
tion shows that A = [~, 0, .... 0] ~ and B = [fi, 0, ..., 0] ~ for some c~, fl ~ GF(p~) • F r o m here on, by the paragraph following Result 1, we will assume that t h e r e exist at least two non-zero indices u 4= v such that a u 4= 0 4: G ; otherwise zt is a generalized twisted field plane; Suppose that t is the line at infinity. Since g transforms the component V(M(y)) onto V M taence - has o r d e r
y f / ~ ' ~ ( p e - 1)(1-pV)
here,
we get that 0 ~ = 1 if and only if (fl-~S= I.
. A l s o by ( 2 ; 1 ) w e have t h a t \a/['-}
--= ~ ,
f o r i = u,v. F r o m
Q!)(p%, - 1} (1 -- pU)
|~}
= cd 1-p')(!-v') =
\~/
181
(p~ - I) (I - p~) mod
(4.1)
a
Thus
( p ~ " - t)(I
-p")=-
~. Therefore,
p..+.+p.,,+p.__p.,,+.+pe.+p,
mod ( ~ )
o
In case that l = O(0) the order of 0 is equal to the order of c~ and if t = O(cc)~ then
171 = I fll. In either case we obtain congruence (4.t), Let m be the mutfiplicative order of p modulo g. Then m [ n Since p"--- i m o d # . Also pC(U)_ 1 m o d # , where q~ is the Euler (p-Nnction, so mlq)(#) and since ~o(N < # (for # > 1) we have that m < n. Hence n = f r o for some integer f > 2. Since # I p '~ - - 1 , then N - P~P"- 11 P " ; (4.2)
1 . So from (4.1) we get
p~+" + p~, + p~ ~ p~,+~ + p~; + p~' m o d N .
Notice that by Result i, e, # u # v and G # G # v. Following the proof of Theorem B in [3] and from the !emmas in [3] [Section 3] we have that if congruence (4.2) holds, then p = 5 and f = 2, or p = 3 and in this case we may have that f = 2, or f = 3 and m = 2, or f = 4 and m = 1. I f # = 3 , f = 4 and m = i, then rc has order 3 4 and go is an autotopism of ~ of order 5, hence n is a generalized
Vol. 63, 1994
Semifield planes with a transitive autotopism group
21
twisted field p l a n e by T h e o r e m 3.1. I n case that p = 3 , f = 3 a n d m = 2, then n = 6 a n d since # I n a n d # l P " - 1 we have that # = 2. H o w e v e r the multiplicative order of p = 3 m o d u l o # ( = 2) is m = 1, a c o n t r a d i c t i o n . If f = 2, then n = 2m, a n d since raft#(#) a n d #[ n we have that #l 2~o(#). H e n c e # = 2% for some integer a. Thus, 2~o(g) = # a n d since nl2qo(#) we have that n - - - # . Therefore a > 2 a n d m = 2 a-1 T h e case p = 5 a n d f = 2 is n o t possible since 52~ - 1 rood 2 a, for a > 2 c o n t r a d i c t ing the definition of re. I f p = 3 a n d f = 2 then, since 3 2a 2 _..= I r o o d 2 ~ for a > 3, we m u s t have that a = 2, so n = # = 4 a n d from T h e o r e m 3.1 it follows that n is a generalized twisted field plane.
Theorem 4.2. Let n be a non-Desarguesian semifield plane of order p", where p + 2 is a prime number and n > 3. Let G be the autotopism group relative to an autotopism triangle A ofn. I f G induces a group 2 transitive on the non-vertex points of a side of A, then n is a generalized twisted field plane, except possibly when p" = 3 6 and G -~ SL(2, 13). P r o o f. Since 2 is transitive o n a set with pn - 1, then I G[ = k(p n - 1), for some integer k. I n case that (i) G < FL(1, p") (see the i n t r o d u c t i o n ) , then since F L ( I , pn) =
(s,t[sP"-l=t"=l,t-lst=s that 121 = k ' 1 2 r ~ ( s ) l ,
p) a n d G c ~ ( s )
~ G ( s ) < FL(I,__p") _~ ( t ) , (s)
=
(s)
where k' divides n. Now, from k ' 1 2 r ~ ( s ) [
we have
= k(p"-1)
we
infer that k ] k' a n d that [ 2 c~ ( s ) [ - p" - 1 , where p = / ~k' is a divisor of n. By T h e o # rem 4.1 the c o n c l u s i o n of this t h e o r e m follows. I n case that (ii) p" = 3 4, then 5 I I G[ so, from T h e o r e m 3.1, n is a generalized twisted field plane. We have the following two corollaries. The second one follows from T h e o r e m A in [5].
Corollary 4.1. Let n be a non-Desarguesian semifield plane of order p" + 3 6, p an odd prime, n > 3. I f n is of rank 3, then n is a generalized twisted field plane.
Corollary 4.2. Let ~ be a translation plane of order pn, p an odd prime, n > 3. I f n admits a collineation group that f i x e s a point L on the line at infinity, l~, and acts 2-transitively on l~ - {L}, then z~ is Desarguesian or a generalized twisted field plane. References [1] A.A. ALBERT, Generalized twisted fields. Pacific. J. Math. 11, 1-8 (1961). [2] M. CORDEROand R. FIGUEROA,Towards a characterization of generalized twisted field planes. J. Geom., to appear. [3] R . FIGUEROA, A characterization of the generalized twisted field planes of characteristic > 5. Geom. Dedicata, to appear. [4] M.J. GANLEYand V. JHA, On a conjecture of Kallaher and Liebler. Geom. Dedicata 21, 277 289 (1986).
22
M. CORDERO and R. F. [;IGUEROA
ARCH. MATH.
[5] M.J. GrANLEYand V. JHA, On transiation planes with a 2-transitive orbit on the iine at infinity Arch. Math. 47, 379-384 (1986). [6] K. ZS~GMONDY,Zur Theorie der Potenzreste. Monatsh. Math. Phys. 3, 265--284 (t892). Eingegangen am 23. 8. !993 *) Anschriflen der Autoren: Minerva Cordero Department of Mathematics Texas Tech University Lubbock, Texas 79409 USA
*) Eine Neufassung ging am 23. 11. 1993 ein.
Raul E Figueroa Department of Mathematics University of Puerto Rico Rio Piedras Puerto Rico 00931
