- Email: [email protected]
S-Partitions of Groups and Steiner Systems
Annals of Discrete Mathematics 30 (1986) 69-84 0 Elsevier Science Publishers B.V. (North-Holland)
S-PARTITIONS OF
69
GROUPS
AN0 STEINER SYSTEMS
Mauro Biliotti Dipartimento di Matematica Universit2 di Lecce Lecce - ITALIA
In this paper we investigate a special class of S-partitions of finite groups. These 5-partitions are used for the construction of resolvable Steiner systems. Several classification theorems are also given. The concept of S-partitions may be traced back to Lingenberg [ 131 although the actual introduction was made by Zappa [ 2 4 ] in 1964. Zappa developed some ideas of Lingenberg so as to provide a group-theoretical description o f linear spaces with a group of automorphisms such that the stabilizer of a line acts transitively on the points of that line. Afterward Zappa [ 2 6 ] and Scarselli [17] mainly investigated the following question: find conditions on a S-partition Z: o f a group G relating the existence o f C to that of a partition - in the usual group-theoretical sense - o f a subgroup o f G. In this case the linear space associated to C is simply the translation AndrC structure associated to that partition [ 3 ] . From a geometrical point of view, the work of Zappa [ 2 5 ] , Rosati [16] and Brenti [6] on the so-called Sylow S-partitions seems to be more interesting as Sylow S-partitions are useful in constructing some classes of Steiner systems. In this connection, another class of S-partitions is noteworthy. These S-partitions are those considered by Lingenberg [ 131 and later bv Zappa [ 2 4 ] , We shall call these S-partitions "Lingenberg S-partitions". Lingenberg S-partitions were inspired by a reconstruction method of the affine geometry A G ( n , K ) , K a field, by means of a special class of subgroups of SL(n,K). In this paper, we study Lingenberg 5-partitions o f finite groups. We mainly investigate "trivial intersection" 5-partitions which we call type I S-partitions (see section 2). For type I S-partitions, we give a "geometric" characterization and somewhat determine the corresponding group structure and action. Also we obtain a classification theorem for Lingenberg S-partitions o f doubly transitive permutation groups. We note that for some simple groups, Lingenberg S-partitions are useful in constructing resolvable Steiner systems. In these cases, the Steiner systems might be regarded as a natural affine geometry for the groups.
70
M . Biliotti
1. PRELIMINARIES
Groups and incidence s t r u c t u r e s considered here are always assumed t o be f i n i t e .
I n general, we s h a l l use standard n o t a t i o n . I f G i s a group and H 2 G, K 9 G, then O(G) i s the maximal normal subgroup o f odd order o f G, S (G) i s t h e s e t o f a l l P
Sylow p-subgroups of G and HK/K i s denoted by A. If H l l K = <1> then K X H denotes the s e m i d i r e c t product of K by H. I f G i s a permutation group on a s e t R and r G f i then G denotes t h e g l o b a l s t a b i l i z e r o f ?i i n G. A s e t R i s a G-set i f t h e r e i s a
r
homomorphism cp from G i n t o the symmetric group on G. Usually we s h a l l w r i t e GR instead o f v(G).
L e t G be a group and S a subgroup of G with SzG. A s e t C o f n o n - t r i v i a l subgroups o f G such t h a t ICl22 i s s a i d t o be a (keguLatr) S-pwrLLtiion o f G i f the f o l l o w i n g c o n d i t i o n s are s a t i s f i e d : (i) (ii) (iii)
S H n s K = S f o r each H , K e Z with H#K;
f o r each g H
4
t
G t h e r e e x i s t s H t C such t h a t g f SH;
C i m p l i e s d-'Hs t C f o r each o E S.
The above d e f i n i t i o n i s due t o Zappa [ 2 4 ] . Here we are i n t e r e s t e d i n t h e f o l l o w i n g special class o f S-partitions: a S - p a r t i t i o n C o f a group G i s s a i d t o be a L i n g e n b a g S . - p a t , t i L i o ~ w l t h respect t o t h e subgroup T o f G i f t h e f o l l o w i n g hold:
(j) T 6 S < NG(T) < G; (jj) G ,
nsx =
xt G
;
(jjj) C = { T * : x e G - N G ( T ) } u { N G ( T ) } . We p o i n t out t h a t C i s determined by the t r i p l e (G,S,T).
Now we g i v e some geome-
t r i c a l definitions. As usual a Lineah npace i s a p a i r ( T I , R ) , where
II i s t h e s e t o f p o i n a and R i s a
f a m i l y o f subsets o f Il whose elements are c a l l e d lines, such t h a t two d i s t i n c t p o i n t s l i e i n e x a c t l y one l i n e . Here we assume a l s o t h a t 11 contains a t l e a s t t h r e e non-collinear points.
(n,R,//), where ( n , R ) i s a l i n e a r is an equivalence r e l a t i o n on R such t h a t each equivalence c l a s s
An AndkL btkuctuhe ( A - n h c t u h e ) i s a t r i p l e
space and
1'//1'
gives a s e t - t h e o r e t i c a l p a r t i t i o n o f Il. L e t 1' = ( I I , R , / / )
be an A-structure and l e t I I o = I I . Assume t h a t whenever P;Q,Rt
no
w i t h P#Q then the l i n e PQ and t h e p a r a l l e l l i n e through R t o PQ are wholly contained i n
no.
I f we s e t R o = { h : h b R , itcIIo}then
(or a l i n e ) and i t i s s a i d t o be a dubspace o f
(no,Ro,//) i s an A - s t r u c t u r e
Y . Isomorphisms and automorphisms
o f l i n e a r spaces and A-structures a r e d e f i n e d i n t h e u s u a l manner with t h e r e q u i r ement t h a t the p a r a l l e l i s m i s preserved f o r A-structures. A h u o t w a b t e SXeinet oybtem w i t h pakametm ( v , f i ) i s an A-structure with v p o i n t s such t h a t each l i n e contains e x a c t l y 12 p o i n t s .
71
Spartitions of Groups ond Steirrer Systems
Following Zappa [24] a q u w i - . t n a ~ h . t i o n A-4tJLuctuhe is an A-structure $=(N,K, / I ) together with a set ( o r pencil) 8 of lines and a family 8 = I T ( & ) : a c 8) of automorphism groups of 0 satisfying the following conditions: (Q,) for each P c IT there exists exactly one line JL t 0 such that P c h ; (GI2) for each n E R there exists exactly one line k E 0 such that c / / 4 ; ( Q 3 ) for each a E o the group T(K) fixes every point of k and every line parallel to t, leaves 0 invariant and acts transitively on the points of each line 4 such that 4 / / @ and b # t . In the following we shall say that (D is a quasi-translation A-structure with respect to the pencil 0 and the family 8. It is very easy to see that, starting from 0 , a new A-structure 5 may be obtained by adding to 0 a new point 0 incident with precisely the lines of 0. 5 will be called the compLetion of 0. Any automorphism a of @ extends to $ by setting a(0) = 0. Lingenberg S-partitions are essentially the geometrical counterpart of quasi-traslation A-structures. If Z is a Lingenberg S-partition of a group G with respect t o the subgroup T, define [G,S,T] as the triple given by - the set of right cosets o f S in G; - the set S ( C ) of the complexes of the form SXg with X E 1, y 6 G ; - the following relation "//" on S(Z): i f a = SN ( T ) q = NG(T)y, then a'//& if and only if either 4 ' = h or a' = STXz G with TX#T and Tg=Txz; if J L = STXy with TX+T, then ~ ' / / Jif L and only if either h ' = N G ( T ) z where TZ=TXY or J L ' = STWz with Tw#T and Txg=Twz. The definition of the parallelism seems to be very involved, but it will make clear in the following. PROPOSITION 1.1 (Zappa [24],2.2). [ 11 t h e
: Sx
mappingh
it4et6
60m
ib0tnOtphiC
( 2 ) [G,S,T]
+
The t G p L e [G,S,T]
Sxy w i t h g E
h a t h e 6 a t t o w i n g pkopek.tiu:
G o d t h e neR 0 6 t i g h t coht& 0 6 5 i n G i n t o a6 autornohphi~mo 06 [G,S,T] w h i c h i b
u point-tJLaMnitiue gkaup
Ro G ;
i n a q u u n i - t t a m t a t i o n A-n,i3uctutuhe
eLemen& a m t h e l i n e n NG(T)x w i t h x
E
w i X h henpect t o
t h e pen& Y
0 whobe
G, and t o the 6 a m i L y 9 = I T x : x t G j .
Obviously in [G,S,T], we have that the point Sx belongs to the line SXg if and only if S x c S X y . Also the converse holds. Indeed, we have: PROPOSITION 1.2 (Zappa [24],2,1). Let 0 be
a quaoi-tkumLaLhn A-Athuctuhe w i t h
rtenpect t o .the penciL 0 und t o t h e t ; a m d g 9 = I T ( & )
: JL 6 El}. Fuhthemohe unnume 3 conjuga.te 4 u b g ~ ~ o u0p5 ~G .- < T ( k ) : J L C 8>. 16 P 0 a p o i n t 06 @, p & t h e Pine 06 8 . t h o u g h P and S = Gp, .then C = {T()L) : a E 8- { p } } LJ u Linyenbety S - p a h R i L o n o d G d R h ~ ~ e n p e oto t T ( p ) and u{NG(T(p))}
2, a cvmpte.te d a b
0
2
IG,S,T(p)l.
06
M. Biliotti
I2
L e t 5 be a q u a s i - t r a n s l a t i o n A-structure w i t h respect t o the p e n c i l 0 and f o r each J L E O l e t H ( h ) denote t h e group o f a l l t h e automorphisms o f 5 f i x i n g every p o i n t o f
and every l i n e p a r a l l e l t o
4.
JI
and which leave 0 i n v a r i a n t . Then @ i s a quasi-
- t r a n s l a t i o n A-structure with respect t o Q and t o t h e f a m i l y 8 = { H ( d : k c O } . Moreover, 9 i s a complete c l a s s o f conjugate subgroups o f G = c H ( h ) : h e @ > so t h a t 5 may always be represented i n t h e form [G,Gp,H(p)l,
2. PROPERTIES AND EXAMPLES
each x,y
t
OF LINGENBERG S-PARTITIONS
d e t e m i n e u Lingenbetlg S - p a k L i t i o n , t h e n TXnTY= 6vtl
LeR (G,S,T)
LEMMA 2.1.
where P E ~ E O .
G w i t h TX#TY. W
N
Paood. Consider t h e A-structure [G,S,T] and l e t ? t T X n T Y . Each p o i n t P of [G,S,T] i s on a p a r a l l e l l i n e t o NG(T)x and a l s o on a p a r a l l e l l i n e t o NG(T)y. Since these l i n e s a r e d i s t i n c t and b o t h are f i x e d by
?
then ?(P) = P and t h e r e f o r e
?
= I.
Lingenberg S - p a r t i t i o n s may be d i v i d e d i n t o two classes according t o t h e f o l l o w i n g definition:
C ad G w i t h kenpect t o t h e bubgtloup T LA 06 t y p e 1 4 6 G w i t h T X # T . ld t h e S - p a h t i L i o n 0 not 06 t y p e I,we 4haU
a Lingenbetlg S-pwc.tLtion SnTX
ench x t it i~ a6 t y p e 11.
<1> d o t
=
bay t h a t
For Lingenberg S - p a r t i t i o n s o f type I we have:
d e t e m i n e n a Lingenbekg S-pakLLtihion 06 .type 1 4 and o d y 7 0 6 [G,S,T] act6 berniheguLahey on t h e Linen 0 6 0 d i b .tinct 6kom NG(T). Fuhthemohe, 4 (G,S,T) d e t e m i n e n a Lingenbekg S-pahtLtion 0 6 t y p e It h e n t h e duUowing hold: ( I ) 7 u c . i ~ t l e g U y on each .!he p a k a f i d t o NG(T) and dinLLnct @om .them, (G,S,T)
PROPOSITION 2.2.
id t h e automotlpkiom gkoup
121 N G ( T ) n T X = <1> 6ok each TK#T, and 131 [NG(T):S]
= ITI-1.
R o o d . Assume (G,S,T)
x
t
determines a Lingenberg S - p a r t i t i o n o f t y p e I and l e t
NG(T) n T y w i t h TY#T. Then
?
f i x e s both t h e l i n e NG(T) and each l i n e p a r a l l e l t o
N (T)q, so t h a t NG(T) i s pointwise f i x e d by G i t f o l l o w s t h a t x = 1 and ( 2 ) holds. Now l e t
w t G w i t h ?#T.
Then
and so z = 1 and
7
Z E
x. Therefore,
x c S and from SnTY=<1>,
z 6 T and NG(T)w,z = NG(T)w f o r some
NG(Tw) and hence z t NG(Tw)nT. But, by (21, N G ( T W ) n T = < l >
a c t s semiregularly on the l i n e s o f 0 d i s t i n c t from NG(T). The
argument may be reversed t o prove t h e converse. L e t k be a l i n e p a r a l l e l t o NG(T) and d i s t i n c t from them and assume ?(R)
= R f o r some
through R, which i s d i s t i n c t from NG(T), i s f i x e d by
on 0
-
{NG(T)},
z t. T,
z.
R 6 a. Then t h e l i n e o f 0
Since
?
acts semiregularly
i t f o l l o w s t h a t z = 1 and ( 1 ) i s proved. Now l e t lGl=g, ING(T)I=n,
I S ( = b and ( T I = t . I n G,
t h e r e e x i s t g/n - 1 d i s t i n c t complexes o f t h e form STX with
Tx#T and each complex contains e x a c t l y n t elements since S n T X = c1>. I f we take
73
S-Partitions of Groups and Steiner Systems SNG(T) = NG(T) c o n t a i n s n elements o f G and (G,S,T)
account o f t h e f a c t tha:
mines a S - p a r t i t i o n of G, then we must have ( n t - n ) ( g / n
- 1)
deter-
n = g so t h a t ( 3 )
t
now f o l l o w s .
16 (G,S,T) d e t e h m i n u a Lingenbefig S-pahtition 06 t y p e It h e n h a h u o l w a b d e SReinek nyotem w i t h paharneteu [ w , k l whehe w = [G:S1 + 1
COROLLARY 2.3. [G,S,TI
and k
=
ITI.
P J L U O ~ .This i s an immediate consequence o f (1) and ( 3 ) o f P r o p o s i t i o n 2.2.
Let 0 be a quabi-Xhun&!ativn A-bthuCtuhe w i t h henpeCt t o t h e pencil 0 and t o t h e damily 8 = { T ( k ) : h e 01. An auhomokpkinm a 0 6 @ c e n t ~ ~ a l i z eT(t) n
PROPOSITION 2.4.
it 6 i x e 4 &Why &ne 0 6 0. A non-idenLicCLe automohpkcnm 0 6 i x i n g evehy f i n e 0 6 0 a c d 6 . p . 6 . on Rhe n e t ofi p o i n d 0 6 0.
d o h each k e 06
0 id and o n d y
46
Phoo6. I f u f i x e s every l i n e o f 0 then, by [24],3.1, a c e n t r a l i z e s T ( k ) f o r each
~ € 0 Conversely, . assume n c e n t r a l i z e s T(h) f o r each J L C O and t h e r e e x i s t s 6 E 0 such t h a t a ( n ) # n . T(o) f i x e s every p o i n t o f n and every l i n e p a r a l l e l t o 4 . L i k e wise, T(n) = a - ’ T ( n ) a f i x e s e v e r y p o i n t o f u ( n ) and every l i n e p a r a l l e l t o u ( n ) . Since n # a ( 4 ) , i t i s easy t o see t h a t t h i s y i e l d s T(n) = <1>, which i s impossible. Now l e t a. be an automorphism o f 0 f i x i n g every l i n e o f 0 and assume a(P) = P f o r
some p o i n t P. I f
JL
i s a l i n e through P and A / / & ,
6 E 0 then u ( b ) =
n and so a ( & ) / / &
which i m p l i e s ~ ( h =) fi. Therefore, a f i x e s every l i n e through P. L e t Q be a p o i n t d i s t i n c t from P and assume PQ=q{ 0. I f w denotes the l i n e o f 0 through Q, we have t h a t a ( Q ) = a ( q f \ w ) = a ( q ) n u ( w ) = q i ? w = Q. I f , on t h e c o n t r a r y , q t 0 then t h e r e l a t i o n a ( Q ) = Q can be obtained by u s i n g t h e same argument as above by s t a r t i n g from a p o i n t P ’ { 4 . The t h e s i s a = I now f o l l o w s . Now we s h a l l g i v e some examples o f Lingenberg S - p a r t i t i o n s . We assume the reader i s acquainted w i t h t h e s t r u c t u r e o f groups SL(2,q); PSU(3,qz), q = ph p a prime; S Z ( ~ ~ ~ ’ R ’ )( 3; 2 n f ’ ) , n > l , and a l s o w i t h the elementary
,
p r o p e r t i e s o f l i n e a r groups. General references are i n [ll] and [12]. I n p a r t i c u l a r , f o r Suzuki groups S Z ( ~ ‘ ~ ” ) , Ree groups R(3“”)
and PSU(3,q’)
see [20], [22] and
[23], [7]r e s p e c t i v e l y . EXAMPLE I. G
2
SL(2,q),
q = ph , q>2. L e t P c S (G) and assume T = S = P; then i t i s P
an easy e x e r c i s e t o show t h a t (G,S,T) I and t h a t [G,S,T],
determines a Lingenberg S - p a r t i t i o n o f type
the completion o f [G,S,T],
i s t h e a f f i n e plane over GF(q).
This i s t h e c l a s s i c a l example which i n s p i r e d t h e work o f Zappa [24]. I t a l s o exp l a i n s the d e f i n i t i o n o f t h e p a r a l l e l i s m i n [G,S,T] EXAMPLE 11. G
7
as given i n s e c t i o n 1.
S z ( q ) , q = Z Z n f ’ , ~ 2 1 .L e t P E S2(G) and l e t Z(P) be t h e c e n t r e o f P.
I f we assume T = Z(P) and S = P then (G,S,T)
determines a Lingenberg S - p a r t i t i o n
o f type I.Indeed, as i t i s w e l l known, NG(T) = N ( P ) and i f x { N G ( T ) then G NG(T)nTX = so t h a t NG(T) n S T X = S. Now l e t g E S T X n S T Y w i t h T # T X # T Y # T , then
M . Biliotti
74 g =
h,.t:
= h2RY w i t h b 1 , n 2 e S, R l , . t t 2 E T and hence
n;'n,
=
d1.t;1 1 x .
If. t , f l , t n f l
and G i s regarded as a c t i n g i n i t s usual doubly t r a n s i t i v e r e p r e s e n t a t i o n o f degree
q z + l then the element .t$(.t;'JX, being t h e product o f two i n v o l u t i o n s w i t h o u t common f i x e d p o i n t s , f i x e s an even number o f p o i n t s . But h;'o1
l i e s i n a Sylow 2-subgroup
o f G and hence i t f i x e s e x a c t l y one p o i n t which i s a c o n t r a d i c t i o n . As we have p r e v i o u s l y shown, we cannot have R =1 f o r only one 4=1,2 and so R =1 for 4=1,2 and 4
~ E S This . y i e l d s S T x n S T Y = S. We =
S t i l l
have I T 1 = q , IS1 = q ' , :NG(T)I
q 2 ( q - I ) , I G / = ( q z + l ) q 2 ( q - I ) and hence, i f T X ' ,
...,TXQ2
=
are t h e q 2 subgroups o f
G which are conjugate t o T and d i s t i n c t f r o m them, i t i s e a s i l y seen t h a t t h e f o l -
lowing r e l a t i o n holds:
4'
1 ( I S T ' ~- ~ IS^)
This proves the a s s e r t i o n .
t
i- I
The completion o f the A-structure [G,S,T]
I N ~ ( T )=I I G ~ .
i s a r e s o l v a b l e S t e i n e r system w i t h
parameters (q(q2-q+I ) , q ) . EYAMPLE 111. G = P X
2
PSU(3,qz), q = 2 h , h > l . L e t Pe S2(G). I t i s w e l l known t h a t NG(P) =
C, where C il c y c l i c o f order ( q 2 - I ) / d with d = ( 3 , q + l ) . Denote by Cl t h e sub( q t i ) / d and s e t T = Z(P), S = P X C,. Then (G,S,T) determines
group o f C of order
a Lingenberg S - p a r t i t i o n o f type I.Indeed, we have again NG(T) = NG(P) and, i f
x f NG(T), N G ( T ) n T X = <1>, so t h a t NG(T)nSTX = 5. Now l e t TX, Ty be such t h a t T#TX+TYgT. By w e l l known p r o p e r t i e s o f G, we have t h a t M = = SL(2,q) and M'ING(P) = <1>or Z ( P ) X C,, w i t h C, c y c l i c o f order q - I . Since q i s even, we have also t h a t ( q - I , q t ] / d ) = I and so, i f S n M # < l > then S n M = Z(P) = T. But, as we
I. (M,T,T) determines a Lingenberg S - p a r t i t i o n o f type I and
have seen i n Example
hence T X T Y n T = c l > . I t f o l l o w s t h a t T X T Y n S = <1>and t h e r e f o r e S T X n S T Y = S. The t h e s i s can now be achieved by a c a l c u l a t i o n s i m i l a r t o t h a t c a r r i e d out i n Examp l e 11. i s a r e s o l v a b l e S t e i n e r system w i t h
The completion o f the A-structure [G,S,T]
I n t h e case q=Zh, with h even, t h i s S t e i n e r system has
parameters ( q ( q 3 - q 2 + I ) , q ) .
been already obtained by Schulz [la]. EXAMPLE I V . G
2
R(q), q=
32n+I
,
~ 2 1 .We s h a l l make use o f t h e r e p r e s e n t a t i o n o f G
i n P G ( 6 , q ) due t o T i t s [22],§5. L e t xI,x2,. ..,x7 be a coordinate system f o r P G ( 6 , q ) 3n+I Furthermore, l e t I be t h e hyperplane o f x + x
and l e t o c A u t ( G F ( q ) ) , a :
.
P G ( 6 , q ) o f equation x , = O and denote by A the a f f i n e space obtained from P G ( 6 , q ) by assuming
I as the i d e a l hyperplane. Then x=x,/x7,
y=x2/x7,
z=x3/x,,
u=xs/x7,
u=xs/x7,
w=xb/x7
i s a non-homogeneous coordinate system for A . F i n a l l y , s e t and denote by
r -
( m ) : ( I ,O,O,O,O,O,O) { ( m ) } the s e t o f p o i n t s o f A whose coordinates s a t i s f y t h e
equations
(1)
- X Z t yo p y a - za f xy' xzo - x o + i y
u = xzy
p + 3
u
t
=
1o =
yz -
X'CJ'3 x2y2
-
- zz
X ~ a f 4
75
S-Partitions of Groups and Steiner Systems Then G
PGL(6,q)r
2
and G a c t s on
r
i n i t s usual doubly t r a n s i t i v e r e p r e s e n t a t i o n
o f degree q 3 + l . L e t P be t h e unique Sylow 3-subgroup o f G l y i n g i n G(,+.
By u s i n g
t h e r e s u l t s o f T i t s [ 2 2 ] , § 5 , about t h e r e p r e s e n t a t i o n o f t h e elements o f P as w e l l as the f a c t t h a t I Z ( P ) I = q , i t i s n o t hard t o prove t h a t t h e p r o j e c t i v i t i e s l y i n g i n Z(P) are e x a c t l y those o f t h e form t c : (XI,xZ,x3,X4,x-,xb,X-)
+
+ ( X I, X L , X 3 + C X 7 , - C X I+ & + , c x ~ + X ~ - C ' X ~ ,C'X1-2CX3+X6-CZX7
c
,X7),
E
GF(r().
According t o T i t s [ 2 2 ] , § 5 , we have a l s o t h a t
: ,xZ,xJ,X4,x5,X6,x7) (X5,xb,X3,x2,xl ?-x7,-X6) i s an i n v o l u t o r i a l p r o j e c t i v i t y o f G which does n o t l i e i n G
fA
(a).
Therefore,
uZ(P)W i s the c e n t r e o f a Sylow 3-subgroup Q o f G which i s d i s t i n c t from P. Now
i t i s our aim t o prove t h a t i f c , d c GF(q), c + O , d#O, then
'""tCi does dd
n o t belong
t o any Sylow 3-subgroup o f G. Since NG(P)nNG(Q) = E, where E i s c y c l i c o f order
q-l
, and
Z(P)E i s a Frobenius group w i t h Frobenius k e r n e l Z(P) (see [23],111.4),
we can suppose, w i t h o u t
loss o f g e n e r a l i t y ,
d=l. We then have
dCdI : (X,,X2,X3,X,,Xj,X6,XI) ( X I + CX, + coxE,x 2 - cx j, ( J + ZC) x g X-'C - ( C + c P ) x 6 + x ,, - x + ( I - c)x, -c"xE,x 2 - 2cx3+( I -C+C~)X,+C'X 6-X7, X,-(2*2C)X3+CX,,'CuXSf
( It2CtC"c')X6-X7
2CX3'CoXg-c2Xgfx7)
A s t r a i g h t f o r w a r d c a l c u l a t i o n shows t h a t d c d Jpossesses the eigenvalue I whose
eigenspace i s generated by t h e v e c t o r ( 0 , I ,l/'Z,-cu~*,O,l/c,l).
Now suppose d c d J
l i e s i n a Sylow 3-subgroup o f G, then t h e f o l l o w i n g hold: - ili.tc does dI n o t have any eigenvalue d i f f e r e n t from I , f o r we are i n charact e r i s t i c 3;
-
LK
C
wx
I
must f i x a p o i n t o f
r
- {
(m)}.
From t h a t which we have proved p r e v i o u s l y , we can i n f e r t h a t t h e f i x e d p o i n t o f
d2dl
on
r -
{ (a)] must have non-homogeneous coordinates ( O , l , I /2, - c O - ~ ,0, I /c)
.
But these coordinates do n o t s a t i s f y (l), a c o n t r a d i c t i o n . Now we may argue as i n the previous examples t o show t h a t i f we s e t T = Z ( P ) and S = P then (G,S,T) determines a Lingenberg 5 - p a r t i t i o n o f type The completion o f t h e A-structure parameters ( q ( q 3 - q '+ I ) ,q 1. EXAMPLE V. G = SL(n,q),
[G,S,T]
I.
i s a r e s o l v a b l e S t e i n e r system w i t h
q = ph , n>3. L e t K = GF(q), V = K" and U a 1-dimensional
subspace o f V. Denote by T ( g , p ) t h e t r a n s v e c t i o n ptHomK(V,K) w i t h p(2) = T(U) =
0,p#O.
y
-f
l-u(l)awhere g c
b of u(b)=gj.
For a f i x e d non-zero v e c t o r
{I, T(b,u) : O # U E HomK(V,K),
V and
U, s e t
Then T(U) i s a subgroup o f G (see [11],11, H i l f s s a t z 6.5). F i n a l l y , denote by S(U) t h e subgroup o f G f i x i n g U pointwise. Then (G,S(U),T(U))
determines a Lingenberg
S - p a r t i t i o n o f type 11. I t i s indeed enough t o observe t h a t t h e A - s t r u c t u r e which i s obtained from t h e a f f i n e space A associated t o V by removing t h e o r i g i n 0
i s a q u a s i - t r a n s l a t i o n A - s t r u c t u r e w i t h respect t o the p e n c i l 0 o f the l i n e s
M.Biliotti
16 through
0 (disregarding
the point
which a r e conjugated t o T(U).
0)and
t o t h e f a m i l y 3 o f t h e subgroups o f G
Furthermore, a t r a n s v e c t i o n o f T(U) with hyperplane
ff f i x e s a l l the l i n e s o f 0 l y i n g i n
H , so t h a t
T(U) i s n o t semiregular on 8 . The
a s s e r t i o n now f o l l o w s from P r o p o s i t i o n s 1.2 and 2.2.
3. FURTHER RESULTS ON LINGENBERG S-PARTITIONS OF TYPE I We w i l l r e q u i r e t h e f o l l o w i n g lemma. LEMMA 3.1.
Let G be one 06 t h e 6oCCowing ghoupn: h
SL(?,q), q = p , p p’Lime, 4 2 4 ; S z ( q ) , q = p2 n f 1 p=2, @ I ; S U O , ~ ~,) q - p
i,, h
PSU(~,~’), q=p
p phime, 4 > 2 , 3 1 ~ 7 ;
, P phime,
~22;
R(q), q=pZn*l, p=3, el; and l e i P be a SgCow p-nubgmup
conditioMn: (I1 /TI - I (21
TnZ(G)
0 4 G. 16
T
a nomat dubgtoup
06
NG(P) b ~ ~ q 4 u 2 g
[NG(P):Tl and =
,
then T = Z(P).
Pmod.
We s h a l l i n v e s t i g a t e t h e v a r i o u s cases separately.
L e t G = sL(z,q),
q s 4 . Assume q i s odd. Then I Z ( G ) ( = 2, ] P I = q , ING(P)I = q ( q - 1 )
= N (P)/Z(G) i s a Frobenius group w i t h Frobenius k e r n e l p. Since T Q N, then G Satz 8.16, we have t h a t e i t h e r 7 < or h P. I n t h e f i r s t case, i t
and
r
by [ l l ] , V ,
f o l l o w s t h a t T < PZ(G). But, T n Z ( G ) = <1>and hence T < P, which i m p l i e s T = since P i s a minimal normal subgroup o f NG(P). I n t h e l a t t e r case, c o n d i t i o n (1) y i e l d s T = P. Since P i s elementary abelian, t h e p r o o f i s achieved. The case q even i s s i m i l a r . L e t G = S z ( q ) . N (P) i s a Frobenius group w i t h Frobenius k e r n e l P, moreover G
/ N G ( P ) / = q 2 ( q - l ) , lP1 = q z ,
I Z ( P ) I = q . We have t h a t e i t h e r T 2 P or T c P. Con-
d i t i o n ( 1 ) i s u n s a t i s f i e d when T 2 P. I f T < P, then e i t h e r T 2 Z(P) or Tn Z(P) =
1
= <1>s i n c e Z(P) i s a minimal normal subgroup o f NG(P). As T Q NG(P), we have t h a t
q-l
IT[-I.
So, i n t h e former
case, i t f o l l o w s T = Z(P) from c o n d i t i o n (1). I n
the l a t t e r case we have P = T X Z(P). However, P/Z(P) i s a b e l i a n and hence P must be a b e l i a n which
is a c o n t r a d i c t i o n .
L e t G = S U ( 3 , q 2 ) , 3 1 q + 7 . We have /Z(G)I = 3 and N (P) = P X C , where / P I = 9’ C i s c y c l i c o f order q2-I and c o n t a i n s Z(G).
-
G
Moreover, I Z ( P ) )
N = NG(P)/Z(P)Z(G) i s a Frobenius group with Frobenius k e r n e l plements isomorphic t o clearly
p
c. Since io N, we must have e i t h e r
i s a minimal normal subgroup o f
and hence
7=
and
= q.
and Frobenius com<
p or
r 2 P. B u t ,
<1> i n t h e f i r s t case.
By c o n d i t i o n ( 2 ) and since ( 3 , q ) = I , we then have t h a t T 5 Z ( P ) and thence T = Z(P)
77
S-Partitions of Groups and Steiner Systems because Z(P) i s a minimal normal subgroup o f NG(P). I n t h e l a t t e r case we cannot have T 2 P by c o n d i t i o n
(l), w h i l e T n Z ( P ) = <1> forces P t o be ( T n P ) Z ( P ) , b u t
as we have seen b e f o r e then P must be a b e l i a n which cannot be t h e case. L e t G = PSU(3,q').
The p r o o f i s s i m i l a r t o t h e previous one.
Let G = R(q).
We have NG(P) = P X C, where ( P I = q 3 and C i s c y c l i c of order q - 1 . = NG(P)/P' i s a FrobenMoreover, Z(P) < P' = @ ( P I , IZ(P)I = q , I P ' I = q 2 . Since i u s group w i t h Frobenius k e r n e l have e i t h e r
7
=
<1> or
7 2 p.
p
(see [23],111.11),
as i n t h e p r e v i o u s cases, we
I n t h e f i r s t case T 5 P I . I f T n Z ( P ) #, then
T 2 Z(P) since Z(P) i s a minimal normal subgroup o f NG(P), b u t by [ 2 3 ] , I I I . 2 , T > Z(P) i m p l i e s T = P ' and c o n d i t i o n (1) i s n o t s a t i s f i e d . So T = Z(P). We can-
n o t have TfiZ(P) = -1> s i n c e if X E PI der 2 q z (see [23],111.2)
-
Z(P) then i t s c e n t r a l i z e r i n NG(P) has or-
and hence I T ( > q , c o n t r a r y t o T S P I . I n t h e l a t t e r ca-
se, we cannot have T n P ' = <1>, since for each t Z(P) < PI (see [23],
[23],111.2)
xE
P - P' we have o ( x ) = 9 and
Theorem). Nevertheless, T n P ' # <1> i m p l i e s I T 1 2 q'
x3 6 (see
and again c o n d i t i o n (1) i s n o t s a t i s f i e d . This completes the p r o o f .
The f o l l o w i n g theorem i s concerned w i t h Lingenberg S - p a r t i t i o n s o f type I i n t h e case o f T being o f even order. THEOREM 3.2.
L e L (G,S,T)
d e t w i n e a ling en be^ S-pcmLiAon
06
even o t d m t h e n one 0 6 t h e doLLowing h u t & : G = O(G)T und T iA a Fhobeniun cornpLement; (u) h ( 6 . 1 ) G 2 SL(2,q), 9.2 , h62; T = S = P w i t h P t S2(G); ( b . 2 ) G 2 SZ(~),q - 2 2ntJ a21; T = Z(P), S = P w i t h P6S2(G); i, ( 6 . 3 ) G 2 PSU(3,qz), q = 2 , h22; T = Z(P) w i t h . P G S2(G) and S = ( q + l l / d , &eke d=13,4+11.
type I.1 6 T ha^
= P X
CI with
IC1I =
Phood. I n t h e A-structure [G,S,T], the p e n c i l o f l i n e s 0 i s a t r a n s i t i v e k s e t 4 = N G ( T ) t 0 then E = Ne(?) and, by P r o p o s i t i o n 2.2, ? a c t s semiJl = r e g u l a r l y on 0 - { a } . Then by [ l o ] , Theorem 2, e i t h e r t h e case ( a ) occurs or 2 S L ( ~ , C ( ) , S Z ( C ( ) , PSU(3,qz), SU(3,q') w i t h q = Z h , h > l . I n the l a t t e r case by [ l o ] , Lemma 3, E a c t s on 0 i n i t s u s u a l doubly t r a n s i t i v e r e p r e s e n t a t i o n o f degree q + l , q 2 + 1 , q 3 t 1 , q 3 + 1 r e s p e c t i v e l y . Then, i t is w e l l known t h a t h = Nc(F) with P"eS2(G) and hence Nc(?) = N c ( B ) . By t a k i n g account o f P r o p o s i t i o n 2.2, we see t h a t 7 s a t i s f i e s c o n d i t i o n s (1) and ( 2 ) o f Lemma 3.1. Therefore ? = Z ( P ) . When E = SL(Z,q), w i t h 10l>l. I f
e
Sz(q)
E
or
PSU(3,qz) we o b t a i n (b.1) - (b.3) i n view o f P r o p o s i t i o n 2.2,(3).
= SU(3,q2) w i t h 31qtJ we have
171
= IZ(p)l =
If
q and hence I ~ l ( = q - l Since . 1Z(G)I=3,
t h i s i m p l i e s t h a t 3 Iq- I by P r o p o s i t i o n 2.4, a c o n t r a d i c t i o n . Therefore, t h e case
G"
= SU(3,q2), w i t h 31q+l, cannot occur.
We p o i n t o u t t h a t Examples
I, I 1 and I11 o f s e c t i o n
2 show t h a t t h e cases (b.11,
(b.2) and (b.3) a c t u a l l y occur. On the c o n t r a r y , i t seems very d i f f i c u l t t o achieve a complete c l a s s i f i c a t i o n o f Lingenberg S - p a r t i t i o n s o f type I i n t h e case ( a ) .
M. Biliotti
78
I n succession, we g i v e some r e s u l t s and examples concerning t h i s case.
LeA (G,S,T)
PROPOSITION 3.3.
d&tehmine u Lingenbehg S - p a d t i o n ud t y p e I und
M-
E induced
a F h o b e u p m u t a t i o n gmup on t h e pen& 0 i n %he A-n&ucRuhe [G,S,Tl, then Rhe ~oUoiu4ngh o l d : h ( I )G = M X T, whehe M A a nonabe14un n p e c b l p-ghoup 0 4 m d u q 2 m t ’ w i t h q = p ,
dume [ T I 2 3.
16
m,hLJ; ( 2 1 IZ(G)I = lZ(M)I = 4,
I T 1 = q + I , S = T,
NG(T) = TZ(M).
P m a 6 . By P r o p o s i t i o n 2.4, Z(E) i s t h e k e r n e l o f t h e r e p r e s e n t a t i o n o f E on 0. Go = E / Z ( c ) and a c t s on 0 as a Frobenius group by our assumptions.
c’
Therefore, Denote by
G
the Frobenius k e r n e l o f
By P r o p o s i t i o n s 2.2 and 2.4,
IT], I i l ) = I
Moreover,
since
= G/Z(G) and l e t M
we have t h a t IZ(G)I
I /TI-1
i G such t h a t M/Z(G) =
M.
and hence ( l T l , l Z ( G ) l ) = l .
i s contained i n a Frobenius complement o f
c.
There-
<1> and MT = M X T. I f x t G then, c l e a r l y , T X C M T and hence G = MT and F = i@ . We have = G 2 NG(T) 2 T, so t h a t T = NG(T) and NG(T) = TZ(G). Set (TI = R , then ( Z ( G ) ( = [NG(T):T] 2 [NG(T):S] = t-I by P r o p o s i t i o n 2.2. Since, fore, T n M =
r
as we have p r e v i o u s l y seen, IZ(G)I
ii
Note t h a t since
group of M with P
I
R - 1 , i t f o l l o w s t h a t / Z ( G ) I d-I and S = T .
i s n i l p o t e n t so i s M (see [11], V.a.7).
$
Z(G) and l e t N = PT. Since (G,S,T)
L e t P be a Sylow p-sub-
determines a Lingenberg
S-partitionof type I, t h e f o l l o w i n g r e l a t i o n holds
It‘
(2)
- $1
!nlRc - I )
+
Rc
c
y1
,
= I N ( and c = ( P n Z ( G ) ( . From (2), i t f o l l o w s t h a t t - l j c since n>tc and
where
hence c=X-I and Z(G) < P. This y i e l d s M = P. Consider the commutators o f t h e form [x,g]
with X E T and g t M-Z(G).
We have [x,g]
and hence [ X , g ]
= X-’(g-’xg)
E
TT’.
Each complex TTg c o n t a i n s e x a c t l y t-J n o n - i d e n t i c a l d i s t i n c t commutators o f t h e form [ x , g ] .
Moreover, i f T 9 # T6 then T T g n T T 6 = T and hence, t h e R-7 commutators
l y i n g i n TTg are d i s t i n c t from those l y i n g i n TT‘. by s e t t i n g [M,T].
(GI
Since
= 6 there e x i s t a t l e a s t ( X - I ) ( i - 7 ) t I
Since [ M , T ]
IITX
: X E G}
2 M and I M I = m ( X - I ) , i t f o l l o w s t h a t [ M , T l
= M.
t h e r e e x i s t s a c h a r a c t e r i s t i c a b e l i a n subgroup A o f M such t h a t A group AT contains e x a c t l y a = [ A : Z ( G ) n A ]
2 (t-I)(a-I)tJ
13.4(b),
121,
then
6
Now suppose Z(G). Then the
,
( t - I ) a . The l a t t e r r e l a b u t t h i s c o n t r a d i c t s a r e s u l t o f Zassenhaus [11],III, Sat2
where a>?. From t h i s i t f o l l o w s t h a t A = [ A , T ] t i o n y i e l d s Z(G) < A ,
=
d i s t i n c t conjugate elements of T. By
using the same argument as before, we have ( t - I ) a 2 / A \ 2 I[A,T]I
I
d i s t i n c t commutators l y i n g i n
and ( A ( =
since IZ(G)) > I. Therefore, a c h a r a c t e r i s t i c a b e l i a n subgroup o f M i s
c e n t r a l i n G. I n conclusion we have proved t h a t :
(I) ( I M I , I T I ) = I , (11) [ M , T l = M, (111)
T c e n t r a l i z e s every c h a r a c t e r i s t i c a b e l i a n subgroup o f M.
By a r e s u l t of Thompson [ L l ] , I I I ,
Satz 13.6, we then have t h a t M i s a nonabelian
79
S-Purtitiorisof G r o i q s arid Steirier Systems
special p-group. Moreover, since Z(M) is a characteristic abelian subgroup of M, phtn. Since a we have that Z(M) = Z(G). Let lZ(G)I = t - 1 = p h , h21, and let l M l Frobenius complement of G/Z(G) has order . t = p h + I , it follows that p h +llp"-l. From so this, we have that ph+ I I pn+ph=ph (pn-h+1 ) and hence ph+ 1 I pn-h- Ph =Ph (Pn-2h- I ph+ 1 1 p n - 2 h - I . Let b E P such that bhSn< Ib+ lih. By iterating the above procedure, it . completes the proof. is easy to prove that b must be even and P ' - ~ ~ - I = O This
.
A Lingenberg S-partition of type I satisfying conditions ( 1 ) and (2) of Proposition 3.3 and its associated A-structure will be called 4peciu.e. An example is given below. EXAMPLE VI. Ue assume the reader is familiar with [14],V,§32. Let ~1 be the projective plane over GF(qz), q=ph , and let p be a hermitian polarity of TI. It is well known that the absolute points and non-absolute lines of p make a Stciner system u with parameters b=q+l and v = q 3 + 1 , which is usually called the d u b b i c d unitul. Moreover the group P(U) consisting o f the projectivities of TI leaving U invariant is isomorphic to PGU(3,qZ). According to Bose [ 5 ] , § 6 , for each absolute line p of [ I , we may define a parallelism among the lines of U as follows: a class of parallel lines consists of a non-absolute line fi through p ( p ) and the non-absolute lines through p ( h ) . Note that p ( p ) E h implies p ( 8 ) 6 p. Therefore, the group T ( h ) consisting o f all ( p ( h ) ,&)-homologies lying in P(U) preserves the parallelism just defined in U , because it fixes the line p . The group T(a) fixes each line i? parallel to h and acts regularly on the points o f L lying in U because T(h) has order 9 t J . Moreover, there exists a unique Sylow p-subgroup M of P ( U ) which fixes p ( p ) and so p itself and acts transitively on the Q' non-absolute lines through p ( p ) . It follows that U - {PI is a quasi-translation A-structure with respect to the pencil 0 of non-absolute lines through p ( p ) and to the family 8 = { T ( h ) : h E O}. It is easily seen that: - G =
= M X T(b),
- T(A) acts semiregularly on o
b E @
;
- [A} ;
- IZ(M)I= q and Z ( M ) consists of all (p(p),p)-elations lying in P(U) and therefore
it fixes every line of 0; scts regularly on 0. From this, it follows that Z ( M ) = Z(G) and G/Z(G) is a Frobenius group. Therefore, (G,T(h),T(h)) determines a special Lingenberg S-partition. - I M I = q 3 and M/Z(M)
now consider case (a) of Theorem 3.2. Assume, hebpecR 20 .the bubghoup T 141 Z 0 u L i ~ g e ~ b e hSg- p ~ ~ h L i . t i 0o6~ G 0 6 .type I ~ i R h w i t h IT I t 3 and (44)G = CT, whefie C 0 bo.&ubi?e and C Q G.
We
be the pencil of lines of [G,S,T]. By Propositions 2.2 and 2.4, we have that and hence Z(c) < ?. Let c/Z(E) be a minimal normal subgroup of G/Z(G) contained in e/Z(c). Since $ Z ( E ) and E/Z(c) acts transitively on 0, then, Let 0
(/TI,JZ(c))) = 1
e
M . Biliotti
80
i s the o r b i t o f
i f 4 = NG(T)C 0 and since
k(E)
h under
i s elementary a b e l i a n f o r
t a r y abelian. I t f o l l o w s now t h a t
?
r,
we have t h a t Is11 > 1 . Moreover,
i s solvable, i t f o l l o w s t h a t
a c t s r e g u l a r l y on
R i n v a r i a n t and a c t s semiregularly on s1 -
LR
R. Moreover, ?R =
[ k } . From t h i s , we i n f e r t h a t
i s elemen-
7
leaves
?ITn
is a
Frobenius group. Now s e t F = , 5, = S n F , No = NG(T)nF, I S o \ = h a , = n o , ( T I = t and I F ( =
IN,(
6.
Since C i s a Lingenberg 5 - p a r t i t i o n o f G o f type I,
the f o l l o w i n g r e l a t i o n holds: (3)
(,t~o
S
- I ) t no 2 6. t-1. On t h e o t h e r hand we have t h a t
- bo)(d/na
From t h i s , i t f o l l o w s t h a t n o / b o t
S
Therefore, n o / b o = t-f and ( 3 ) h o l d s as an e q u a l i t y . Using t h i s ,
[N:S] = . t - 1 .
i t i s n o t d i f f i c u l t t o see t h a t ifwe s e t R = : x t F 1 } u {NFl(T1)
then C I = { T f
[N,:S,]
respect t o the subgroup TI.
But
GFs;,
s l =SJR,
T ~ IRA, =
F,= F/R,
i s a Lingenberg S - p a r t i t i o n o f F 1 o f type I w i t h
-R
-R-R
F I = I- T
and hence, by P r o p o s i t i o n 3.3,
XI i s a
b p e c i d lingenbekg S - p a t L i L i u n . From a geometrical p o i n t o f view, t h e previous r e s u l t can be expressed as f o l l o w s .
PROPOSITION 3.4.
L e t C be. a Lingenbehg S-pa/ttLtitian a6 G
0 6 type I
w c t h k e ~ p e c tt o
-the hubghaup -i w i t h T 2 3. Adbume G = CT, whem C i h a hoLwabLe n u m d hubghoup
cuntaim a bubbpace w h i c h i~ a b p e c i d
ud G. Then t h e A-bRhuctwle [G,S,T]
A-btkuc-
tuke.
Pkoo6. L e t [FI,S1,T1]
be the s p e c i a l A-structure r e l a t e d t o the s p e c i a l S , - p a r t i -
t i o n & o f Fi described above. I f S I X i s a p o i n t o f [F,,Sl,T1] the map from t h e s e t o f p o i n t s o f [FL,SI,T1] defined as f o l l o w s
n
-
: SIX
-t
sx
and
x = Kx,
l e t q be
i n t o t h e s e t o f p o i n t s o f [G,S,T]
.
I t i s s t r a i g h t f o r w a r d t o show t h a t 11 i s w e l l d e f i n e d and g i v e s an embedding o f [FI,Sl,TIl
i n t o [G,S,Tl.
4 . LINGENBERG 5-PARTITIONS OF DOUBLY TRANSITIVE PERMUTATION GROUPS determines a Lingenberg S - p a r t i t i o n . Since i n Examples I - V vie have
Assume ( G , I , T ) that: (a)
t h r gnoup
cicib
Z-tcanoiLivek?y un t h e pen&
06 f i n e d
0
06
.the A - ~ i h u c . t u c e
[G,S,Tl. then the n a t u r a l question a r i s e s whether i t i s p o s s i b l e t o c l a s s i f y a l l the t I i p l e s
(G,S,T) which determine Lingenberg S - p a r t i t i o n s s a t i s f y i n g c o n d i t i o n ( a ) . I n the f o l l o w i i i g , we s n a l l prove t h a t a r a t h e r s a t i s f a c t o r y answer t o t h i s question may be g i v e n provided t h a t t h e c l a s s i f i c a t i o n o f doubly t r a n s i t i v e permutation groups i s assumed.
As i t i s w e l l knowri, such a c l a s s i f i c a t i o n f o l l o w s f r o m t h a t o f F i n i t e
simple groups.
(G,S,l) deXehmAneb a Lingenbehg S-pamLCi.on C &Lion ( u l . 16 C 0 ad t y p e It h e n une 06 t h e ,joXCawing h d h : THEOREM 4.1.
Annwrie
b a t ~ A 6 y ~ ncung
81
S-Partitionsof Groups and Steiner Systems
(I]
q = p c l , ph22; T = S = P l ~ i h i hP E S (C); P 2n+ I ntl; T = Z(P), S = P wi2h P 6 S 2 ( G ) ; G 2 Sz(q), Q = G 2 PSU(3,qz), q=2', h t 2 ; T = Z(P) wiRh PE S2(G), S = P A C I , wh&he I C I = ( q + l ) / d ,d=(3,y+ll; R ( q ) , q = 3 2 n C 7 , n 2 i ; T = Z(P), S = P w a h PES3(G). G 0 0 6 t y p e I1 lhen PSL(n,q) S G/Z(G) S PTL(n,q) iyhehe n23. SL(Z,q),
G
(21 (31
(4) 18 C
a LinyenbErg 5 - p a r t i t i o n o f type - determines s a t i s f i e s the following condition:
I. By P r o p o s i t i o n
Pk006. Assume (G,S,T) -,@
t l i e group G
2.7,
* G/Z(E)
(h) f o r each h e @ , t h e s t a b i l i z e r o f s e m i r e g u l a r l y on
o-
5
6'
in
I=
c o n t a i n s a normal subgroup which a c t s
{t}.
From t h e c l a s s i f i c a t i o n theorem o f f i n i t e doubly t r a n s i t i v e permutation groups, we
(see [ 4 ] ) i s a c t u a l l y a theorem (see
h a ? t h a t t h e so c a l l e d "Hering conjecture"
[19],p.302)
c0
asserting that i f
i s 2 - t r a n s i t i v e on 0 and s a t i s f i e s ( h ) then one
o f t h e f o l l o w i n g holds:
c0
(j) c o n t a i n s a r e g u l a r normal subgroup, (jj) Eo PSL(Z,(i), q L 4 , Sz(q), PSU(3,q"), q > 2 , or R ( y ) , 4'3, -1
aiid
Go
a c t s on 0 i n
i t s usual doubly t r a n s i t i v e r e p r e s e n t a t i o n . We s h a l l i n v e s t i g a t e these cases separately.
( j ) . Let
C~c.14
hence
Go =
E0
be t h e r e g u l a r rlorrnal subgroup o f
cO. We have t h a t Go
" G Z ( e ) i s a Frobenius group. If( T I = 2 then
then by P r o p o s i t i o n 3.3,
-,o
we must have I N
I
=
EB =
and
=
= SL<2,2). I f ( T I 1 3
q Z m , where q i s a prime power and m 2 1
tdureoveL, ( T I = q + ~ Since . Go i s 2 - t r a n s i t i v e on
B, i t
iollows that q2m-l=q+l.
T h i s i m p l i e s q = 2 , m = l and i t i s very easy t o show t h a t G = SL(2,3). Cane ( j j ) . Note t h a t i f L 5 G and LZ(G) = G then L = G s i n c e ( \ T l , l Z ( G ) l ) = I and G i s generated by T and i t s conjugates. Therefore, G i s a c e n t r a l i r r e d u c i b l e
tension o f
$).
[11],[12]) then t h e r e e x i s t s a unique r e p r e s e n t a t i o n group H o f of Z(H). Moreover, Z(G) = Z(H)/Z,. I f M(E')
f o r some subgroup Z, multiplier o f 2.2,
ex-
Since, i n t h e case under c o n s i d e r a t i o n , G i s a simple group ( s e e
EB then
I
IZ(H)I = IM(CO)I ( s e e [21],Ch.2,59).
we have t h a t ( Z ( G ) I
assume :
IT1
- I,
i t f o l l o w s t h a t IZ(G)I
I
? and
G = H/Z,
denotes t h e Schur
Since, by P r o p o s i t i o n ( l M ( ~ O ) ~ , ~ T l - JNow ).
q = ph , h 2 2 . I f p i s odd then we cannot have G PSL(2,q) s h c e i n t h i s case every normal subgroup o f NG(P), where P e S (C), c o n t a i n s P and hence r e P l a t i o n ( 3 ) o f P r o p o s i t i o n 2.2 cannot be s a t i s f i e d f o r \ P I = q and ING(P)I = % q [ q - I I . Therefore, when q:J,9 by [11],V,25.7, we must have G = H = SL(2,q) and t h e t n e s i s f o l l o w s from Lemma 3.1. If9 - 4 then T has even order and, by Theorem 3.2, G = = S L ( 2 , 4 ) . L e t q - 9 . Since NG(T)/Z(G) is a Frobenius group o f order 9.4 with Fro-
'"G
= PSL(p,(i),
benius k e r n e l o f order 9 , i t f o l l o w s t h a t I T 1 I 9 . On t h e o t h e r hand,
-
where
101= I0 and hence IT
fore,
/ Z ( G ) I = 2 and G
I
= 9 . By [8],Table 1, we have t h a t M(Co)
SL(2,9).
IT1
= Z6.
I
I01-J There-
82
M . Riliotti
Let
Go --
Sz(g). I n t h i s case T has even order and (2) f o l l o w s from Theorem 3.2.
to
= PSU(3,q2), q > 2 . By [ 9 ] , Theorem 2, we have e i t h e r G 2 PSU(3,q') o r G 2 SU(3,q2), 3 1 q + l , may be excluded by arguing as i n the = SU(3,q'). The case G h p r o o f o f Theorem 3.2. Assume G ^1 PSU(3,qZ), y = p , p an odd prime. By Lemma 3.1, we Let
have t h a t T = Z(P) with P t S (G) and S = P X
P
C1,
where C I i s c y c l i c of order
g+J/d,
d=(3,q+J). Moreover C , < C, where C i s c y c l i c o f order q2-J/d and C < NG(P). L e t C2 be t h e subgroup o f C of order q - J . As i t i s w e l l known, t h e r e e x i s t s a subgroup M o f G such t h a t MnNG(P) = TC2 and SL(2,q)
= M = for a s u i t a b l e conjugate
Let b = = .ti, where I f - t e T and i is t h e unique i n v o l u t i o n i n C2. As was shown i n Example I,
subgroup Tx o f T, where TX may be chosen i n such a way t h a t Cz < NM(Tx). determines a Lingenberg S - p a r t i t i o n o f type
(M,TX,TX)
we have t h a t b
b
E
{ NW(TX). Therefore, t h e r e e x i s t s
I.
Since NM(T)nNM(TX) = C 2 ,
Ty < M with T#TY#Tx such t h a t
TXTY and hence T x T y n S # < l > since b E S. Note t h a t
i E Ci
f o r q odd. I t f o l l o w s
does n o t determine a Lingenberg S - p a r t i t i o n . The t h e s i s can now be
t h a t (G,S,T)
achieved by using Lemma 3.1. Let
Go
R ( g ) , q>3. By [l], Theorem 1, we have t h a t M(?)
= clz and ( 4 ) again f o l -
lows from Lemma 3.1. Now assume (G,S,T)
--
determines a Lingenberg S - p a r t i t i o n o f t y p e I I . I f NG(T)x i s a
l i n e o f 0 then the s t a b i l i z e r NE(T"X)' o f t h i s l i n e i n group Tx'
Eo
c o n t a i n s the normal sub-
s a t i s f y i n g the c o n d i t i o n
(1) yX0f3?@= f o r each
?@such
that
T"yo#T"xxo.
Indeed, assume ( 1 ) does n o t hold. Then TXZ(G)i7TYZ(G) > Z(G) f o r some Ty with TX# fTy.
I t f o l l o w s t h a t T X T Y n Z ( G ) # < l >since TXnTY= by Lemma 2.1. So t h e r e ex-
sists z
E
Z(G),
= S then z
E
-1
~ $ 1 ,such t h a t z c T T Y X E STY'-'.
Since z
E
NG(T) and STY'-'
n NG(T)=
5, b u t t h i s i s a c o n t r a d i c t i o n because Sn Z(G) = <1>. Moreover, by
P r o p o s i t i o n 2.2, we have:
(2)
' 3
does n o t a c t semiregularly on Q - {NG(T)x].
By a w e l l known r e s u l t o f O'Nan [ 1 5 ] , PSL(n,q) 5
6 PrL(n,q)
Theorem A,
c o n d i t i o n s (1) and ( 2 ) i m p l y
with n23. So t h e t h e s i s f o l l o w s from P r o p o s i t i o n 2.4.
As a f i n a l remark, we note t h a t t h e c l a s s i f i c a t i o n theorem o f doubly t r a n s i t i v e permutation groups i s r e q u i r e d o n l y when (G,S,T)
determines a Lingenberg S - p a r t i t i o n
o f type I and T has odd order.
REFERENCES.
[l] A l p e r i n , J.L. and Gorenstein, D., The m u l t i p l i c a t o r s o f c e r t a i n simple groups, Proc. Am. Math. SOC. 17 (1966), 515-519. [ 2 ] Andre, J., Uber P a r a l l e l s t r u k t u r e n , T e i l I : Gundbegriffe, Math. Z. 76 (1961),
85-102.
S-Partitions of Groups and Steirier Systems
[ 31
Andr6, J.,
83
Uber P a r a l l e l s t r u k t u r e n , T e i l I 1 : T r a n s l a t i o n s s t r u k t u r e n , Math. Z.
76 (1961), 155-163.
[41 Aschbacher, M., F-sets and permutation groups, J. Algebra 30 (1974), 400-416. [5] Bose, R.C., On t h e a p p l i c a t i o n o f f i n i t e p r o j e c t i v e geometry f o r d e r i v i n g a
c e r t a i n s e r i e s o f balanced Kirkman arrangements, in:The Golden Jub. Comm., C a l c u t t a Math. SOC. (1958-591,341-354. [6] B r e n t i , F., S u l l e S - p a r t i z i o n i d i Sylow i n alcune c l a s s i d i g r u p p i f i n i t i , Boll. Un. Mat. I t . (6) 3-8 (1984), 665-685. [ 71 Burkhardt, R., Uber d i e Zerlegungszahlen der u n i t a r e n Gruppen PSU(3,2 2 f 1 , J. Algebra 61 (19791, 548-581. [ 8 ] Griess, R.L.,Jr., Schur M u l t i p l i e r s o f t h e known f i n i t e simple groups, B u l l . Am. Math. Soc. 78 (19721, 68-71. Schur M u l t i p l i e r s o f f i n i t e simple groups o f L i e type, [9] Griess, R.L.,Jr., Trans. Am. Math. SOC. 183 (1973), 355-421. [ 101 Hering, C., On subgroups w i t h t r i v i a l normalizer i n t e r s e c t i o n , J. Algebra 20
(19721%622-629. [ll] Huppert, B., Endliche Gruppen I(Springer-Verlag, Berlin-Heidelberg-New York, 1979). [ 121 Huppert, B. and Blackburn, N., F i n i t e Groups I11 (Springer-Verlag, B e r l i n -Heidelberg-New York, 1982). [ 131 Lingenberg, R. , Uber Gruppen p r o j e c t i v e r K o l l i n e a t i o n e n , whelche e i n e perspect i v e D u a l i t a t i n v a r i a n t lassen, Arch. Math. 13 (1962), 385-400. [ 141 Luneburg , H., T r a n s l a t i o n Planes (Springer-Verlag, Berlin-Heidelberg-New York, 1980). [ 151 O'Nan, M.E., Normal s t r u c t u r e o f t h e one-point s t a b i l i z e r o f a doubly-trans i t i v e permutation group. I, Trans. Am. Math. SOC. 214 (19751, 1-42. [16] Rosati, L.A., S u l l e S - p a r t i z i o n i n e i g r u p p i non a b e l i a n i d ' o r d i n e pq, Rend. Sem. Mat. Univ. Padova 38 (19671, 108-117. [17] S c a r s e l l i , A., S u l l e S - p a r t i z i o n i r e g o l a r i d i un gruppo f i n i t o , A t t i ACC. Naz. L i n c e i , Rend C1. S c i . F i s . Mat. Nat. ( 8 ; 62 (1977), 300-304. [18] Schulz, R.H., Zur Geometrie der PSU(3,q ) , i n : B e i t r a g e zur Geometr. Algebra, Proc. Symp. Duisburg, 1976 (Birkhauser, Basel, 1977), 293-298. [19] Shult, E.E., Permutation groups with few f i x e d p o i n t s , i n : Geometry - von S t a u d t ' s P o i n t o f View, Proc. NATO Adv. Study I n s t . Bad Windsheim, 1980 ( 0 . R e i d e l P.C., Oordrecht, 19811, 275-311. [20] Suzuki, M., On a c l a s s o f doubly t r a n s i t i v e groups, Ann. Math. 75 (19621, 104-145. [ 211 Suzuki, M., Group Theory I (Springer-Verlag, Berlin-Heidelberg-New York, 1982) [22] T i t s , J., Les groupes simples de Suzuki e t de Ree, i n : Sem. Bourbaki, 13e annCe, 210 (1960/61), 1-18. 1231 ~- Ward, H.N., On Reels s e r i e s o f simple groups, Trans. Am. Math. SOC. 121 (19661, 62-89. 1241 ZaDOa. G.. S u a l i sDazi a e n e r a l i quasi d i t r a s l a z i o n e , Le Matematiche (Catan i a ) i9 (i9647, 127-143: S u l k S - p a r t i z i o n i d i un gruppo f i n i t o , Ann. Mat. Pura Appl. (4) 1251 . . Zappa, G., 74 (19661, 1-14. [26] Zappa, G., P a r t i z i o n i g e n e r a l i z z a t e n e i gruppi, i n : C o l l . I n t . Teorie Comb. 1973 (Acc. Naz. L i n c e i , Roma 1976), 433-437. i
,
S-PARTITIONS OF
69
GROUPS
AN0 STEINER SYSTEMS
Mauro Biliotti Dipartimento di Matematica Universit2 di Lecce Lecce - ITALIA
In this paper we investigate a special class of S-partitions of finite groups. These 5-partitions are used for the construction of resolvable Steiner systems. Several classification theorems are also given. The concept of S-partitions may be traced back to Lingenberg [ 131 although the actual introduction was made by Zappa [ 2 4 ] in 1964. Zappa developed some ideas of Lingenberg so as to provide a group-theoretical description o f linear spaces with a group of automorphisms such that the stabilizer of a line acts transitively on the points of that line. Afterward Zappa [ 2 6 ] and Scarselli [17] mainly investigated the following question: find conditions on a S-partition Z: o f a group G relating the existence o f C to that of a partition - in the usual group-theoretical sense - o f a subgroup o f G. In this case the linear space associated to C is simply the translation AndrC structure associated to that partition [ 3 ] . From a geometrical point of view, the work of Zappa [ 2 5 ] , Rosati [16] and Brenti [6] on the so-called Sylow S-partitions seems to be more interesting as Sylow S-partitions are useful in constructing some classes of Steiner systems. In this connection, another class of S-partitions is noteworthy. These S-partitions are those considered by Lingenberg [ 131 and later bv Zappa [ 2 4 ] , We shall call these S-partitions "Lingenberg S-partitions". Lingenberg S-partitions were inspired by a reconstruction method of the affine geometry A G ( n , K ) , K a field, by means of a special class of subgroups of SL(n,K). In this paper, we study Lingenberg 5-partitions o f finite groups. We mainly investigate "trivial intersection" 5-partitions which we call type I S-partitions (see section 2). For type I S-partitions, we give a "geometric" characterization and somewhat determine the corresponding group structure and action. Also we obtain a classification theorem for Lingenberg S-partitions o f doubly transitive permutation groups. We note that for some simple groups, Lingenberg S-partitions are useful in constructing resolvable Steiner systems. In these cases, the Steiner systems might be regarded as a natural affine geometry for the groups.
70
M . Biliotti
1. PRELIMINARIES
Groups and incidence s t r u c t u r e s considered here are always assumed t o be f i n i t e .
I n general, we s h a l l use standard n o t a t i o n . I f G i s a group and H 2 G, K 9 G, then O(G) i s the maximal normal subgroup o f odd order o f G, S (G) i s t h e s e t o f a l l P
Sylow p-subgroups of G and HK/K i s denoted by A. If H l l K = <1> then K X H denotes the s e m i d i r e c t product of K by H. I f G i s a permutation group on a s e t R and r G f i then G denotes t h e g l o b a l s t a b i l i z e r o f ?i i n G. A s e t R i s a G-set i f t h e r e i s a
r
homomorphism cp from G i n t o the symmetric group on G. Usually we s h a l l w r i t e GR instead o f v(G).
L e t G be a group and S a subgroup of G with SzG. A s e t C o f n o n - t r i v i a l subgroups o f G such t h a t ICl22 i s s a i d t o be a (keguLatr) S-pwrLLtiion o f G i f the f o l l o w i n g c o n d i t i o n s are s a t i s f i e d : (i) (ii) (iii)
S H n s K = S f o r each H , K e Z with H#K;
f o r each g H
4
t
G t h e r e e x i s t s H t C such t h a t g f SH;
C i m p l i e s d-'Hs t C f o r each o E S.
The above d e f i n i t i o n i s due t o Zappa [ 2 4 ] . Here we are i n t e r e s t e d i n t h e f o l l o w i n g special class o f S-partitions: a S - p a r t i t i o n C o f a group G i s s a i d t o be a L i n g e n b a g S . - p a t , t i L i o ~ w l t h respect t o t h e subgroup T o f G i f t h e f o l l o w i n g hold:
(j) T 6 S < NG(T) < G; (jj)
nsx =
xt G
;
(jjj) C = { T * : x e G - N G ( T ) } u { N G ( T ) } . We p o i n t out t h a t C i s determined by the t r i p l e (G,S,T).
Now we g i v e some geome-
t r i c a l definitions. As usual a Lineah npace i s a p a i r ( T I , R ) , where
II i s t h e s e t o f p o i n a and R i s a
f a m i l y o f subsets o f Il whose elements are c a l l e d lines, such t h a t two d i s t i n c t p o i n t s l i e i n e x a c t l y one l i n e . Here we assume a l s o t h a t 11 contains a t l e a s t t h r e e non-collinear points.
(n,R,//), where ( n , R ) i s a l i n e a r is an equivalence r e l a t i o n on R such t h a t each equivalence c l a s s
An AndkL btkuctuhe ( A - n h c t u h e ) i s a t r i p l e
space and
1'//1'
gives a s e t - t h e o r e t i c a l p a r t i t i o n o f Il. L e t 1' = ( I I , R , / / )
be an A-structure and l e t I I o = I I . Assume t h a t whenever P;Q,Rt
no
w i t h P#Q then the l i n e PQ and t h e p a r a l l e l l i n e through R t o PQ are wholly contained i n
no.
I f we s e t R o = { h : h b R , itcIIo}then
(or a l i n e ) and i t i s s a i d t o be a dubspace o f
(no,Ro,//) i s an A - s t r u c t u r e
Y . Isomorphisms and automorphisms
o f l i n e a r spaces and A-structures a r e d e f i n e d i n t h e u s u a l manner with t h e r e q u i r ement t h a t the p a r a l l e l i s m i s preserved f o r A-structures. A h u o t w a b t e SXeinet oybtem w i t h pakametm ( v , f i ) i s an A-structure with v p o i n t s such t h a t each l i n e contains e x a c t l y 12 p o i n t s .
71
Spartitions of Groups ond Steirrer Systems
Following Zappa [24] a q u w i - . t n a ~ h . t i o n A-4tJLuctuhe is an A-structure $=(N,K, / I ) together with a set ( o r pencil) 8 of lines and a family 8 = I T ( & ) : a c 8) of automorphism groups of 0 satisfying the following conditions: (Q,) for each P c IT there exists exactly one line JL t 0 such that P c h ; (GI2) for each n E R there exists exactly one line k E 0 such that c / / 4 ; ( Q 3 ) for each a E o the group T(K) fixes every point of k and every line parallel to t, leaves 0 invariant and acts transitively on the points of each line 4 such that 4 / / @ and b # t . In the following we shall say that (D is a quasi-translation A-structure with respect to the pencil 0 and the family 8. It is very easy to see that, starting from 0 , a new A-structure 5 may be obtained by adding to 0 a new point 0 incident with precisely the lines of 0. 5 will be called the compLetion of 0. Any automorphism a of @ extends to $ by setting a(0) = 0. Lingenberg S-partitions are essentially the geometrical counterpart of quasi-traslation A-structures. If Z is a Lingenberg S-partition of a group G with respect t o the subgroup T, define [G,S,T] as the triple given by - the set of right cosets o f S in G; - the set S ( C ) of the complexes of the form SXg with X E 1, y 6 G ; - the following relation "//" on S(Z): i f a = SN ( T ) q = NG(T)y, then a'//& if and only if either 4 ' = h or a' = STXz G with TX#T and Tg=Txz; if J L = STXy with TX+T, then ~ ' / / Jif L and only if either h ' = N G ( T ) z where TZ=TXY or J L ' = STWz with Tw#T and Txg=Twz. The definition of the parallelism seems to be very involved, but it will make clear in the following. PROPOSITION 1.1 (Zappa [24],2.2). [ 11 t h e
: Sx
mappingh
it4et6
60m
ib0tnOtphiC
( 2 ) [G,S,T]
+
The t G p L e [G,S,T]
Sxy w i t h g E
h a t h e 6 a t t o w i n g pkopek.tiu:
G o d t h e neR 0 6 t i g h t coht& 0 6 5 i n G i n t o a6 autornohphi~mo 06 [G,S,T] w h i c h i b
u point-tJLaMnitiue gkaup
Ro G ;
i n a q u u n i - t t a m t a t i o n A-n,i3uctutuhe
eLemen& a m t h e l i n e n NG(T)x w i t h x
E
w i X h henpect t o
t h e pen& Y
0 whobe
G, and t o the 6 a m i L y 9 = I T x : x t G j .
Obviously in [G,S,T], we have that the point Sx belongs to the line SXg if and only if S x c S X y . Also the converse holds. Indeed, we have: PROPOSITION 1.2 (Zappa [24],2,1). Let 0 be
a quaoi-tkumLaLhn A-Athuctuhe w i t h
rtenpect t o .the penciL 0 und t o t h e t ; a m d g 9 = I T ( & )
: JL 6 El}. Fuhthemohe unnume 3 conjuga.te 4 u b g ~ ~ o u0p5 ~G .- < T ( k ) : J L C 8>. 16 P 0 a p o i n t 06 @, p & t h e Pine 06 8 . t h o u g h P and S = Gp, .then C = {T()L) : a E 8- { p } } LJ u Linyenbety S - p a h R i L o n o d G d R h ~ ~ e n p e oto t T ( p ) and u{NG(T(p))}
2, a cvmpte.te d a b
0
2
IG,S,T(p)l.
06
M. Biliotti
I2
L e t 5 be a q u a s i - t r a n s l a t i o n A-structure w i t h respect t o the p e n c i l 0 and f o r each J L E O l e t H ( h ) denote t h e group o f a l l t h e automorphisms o f 5 f i x i n g every p o i n t o f
and every l i n e p a r a l l e l t o
4.
JI
and which leave 0 i n v a r i a n t . Then @ i s a quasi-
- t r a n s l a t i o n A-structure with respect t o Q and t o t h e f a m i l y 8 = { H ( d : k c O } . Moreover, 9 i s a complete c l a s s o f conjugate subgroups o f G = c H ( h ) : h e @ > so t h a t 5 may always be represented i n t h e form [G,Gp,H(p)l,
2. PROPERTIES AND EXAMPLES
each x,y
t
OF LINGENBERG S-PARTITIONS
d e t e m i n e u Lingenbetlg S - p a k L i t i o n , t h e n TXnTY=
LeR (G,S,T)
LEMMA 2.1.
where P E ~ E O .
G w i t h TX#TY. W
N
Paood. Consider t h e A-structure [G,S,T] and l e t ? t T X n T Y . Each p o i n t P of [G,S,T] i s on a p a r a l l e l l i n e t o NG(T)x and a l s o on a p a r a l l e l l i n e t o NG(T)y. Since these l i n e s a r e d i s t i n c t and b o t h are f i x e d by
?
then ?(P) = P and t h e r e f o r e
?
= I.
Lingenberg S - p a r t i t i o n s may be d i v i d e d i n t o two classes according t o t h e f o l l o w i n g definition:
C ad G w i t h kenpect t o t h e bubgtloup T LA 06 t y p e 1 4 6 G w i t h T X # T . ld t h e S - p a h t i L i o n 0 not 06 t y p e I,we 4haU
a Lingenbetlg S-pwc.tLtion SnTX
ench x t it i~ a6 t y p e 11.
<1> d o t
=
bay t h a t
For Lingenberg S - p a r t i t i o n s o f type I we have:
d e t e m i n e n a Lingenbekg S-pakLLtihion 06 .type 1 4 and o d y 7 0 6 [G,S,T] act6 berniheguLahey on t h e Linen 0 6 0 d i b .tinct 6kom NG(T). Fuhthemohe, 4 (G,S,T) d e t e m i n e n a Lingenbekg S-pahtLtion 0 6 t y p e It h e n t h e duUowing hold: ( I ) 7 u c . i ~ t l e g U y on each .!he p a k a f i d t o NG(T) and dinLLnct @om .them, (G,S,T)
PROPOSITION 2.2.
id t h e automotlpkiom gkoup
121 N G ( T ) n T X = <1> 6ok each TK#T, and 131 [NG(T):S]
= ITI-1.
R o o d . Assume (G,S,T)
x
t
determines a Lingenberg S - p a r t i t i o n o f t y p e I and l e t
NG(T) n T y w i t h TY#T. Then
?
f i x e s both t h e l i n e NG(T) and each l i n e p a r a l l e l t o
N (T)q, so t h a t NG(T) i s pointwise f i x e d by G i t f o l l o w s t h a t x = 1 and ( 2 ) holds. Now l e t
w t G w i t h ?#T.
Then
and so z = 1 and
7
Z E
x. Therefore,
x c S and from SnTY=<1>,
z 6 T and NG(T)w,z = NG(T)w f o r some
NG(Tw) and hence z t NG(Tw)nT. But, by (21, N G ( T W ) n T = < l >
a c t s semiregularly on the l i n e s o f 0 d i s t i n c t from NG(T). The
argument may be reversed t o prove t h e converse. L e t k be a l i n e p a r a l l e l t o NG(T) and d i s t i n c t from them and assume ?(R)
= R f o r some
through R, which i s d i s t i n c t from NG(T), i s f i x e d by
on 0
-
{NG(T)},
z t. T,
z.
R 6 a. Then t h e l i n e o f 0
Since
?
acts semiregularly
i t f o l l o w s t h a t z = 1 and ( 1 ) i s proved. Now l e t lGl=g, ING(T)I=n,
I S ( = b and ( T I = t . I n G,
t h e r e e x i s t g/n - 1 d i s t i n c t complexes o f t h e form STX with
Tx#T and each complex contains e x a c t l y n t elements since S n T X = c1>. I f we take
73
S-Partitions of Groups and Steiner Systems SNG(T) = NG(T) c o n t a i n s n elements o f G and (G,S,T)
account o f t h e f a c t tha:
mines a S - p a r t i t i o n of G, then we must have ( n t - n ) ( g / n
- 1)
deter-
n = g so t h a t ( 3 )
t
now f o l l o w s .
16 (G,S,T) d e t e h m i n u a Lingenbefig S-pahtition 06 t y p e It h e n h a h u o l w a b d e SReinek nyotem w i t h paharneteu [ w , k l whehe w = [G:S1 + 1
COROLLARY 2.3. [G,S,TI
and k
=
ITI.
P J L U O ~ .This i s an immediate consequence o f (1) and ( 3 ) o f P r o p o s i t i o n 2.2.
Let 0 be a quabi-Xhun&!ativn A-bthuCtuhe w i t h henpeCt t o t h e pencil 0 and t o t h e damily 8 = { T ( k ) : h e 01. An auhomokpkinm a 0 6 @ c e n t ~ ~ a l i z eT(t) n
PROPOSITION 2.4.
it 6 i x e 4 &Why &ne 0 6 0. A non-idenLicCLe automohpkcnm 0 6 i x i n g evehy f i n e 0 6 0 a c d 6 . p . 6 . on Rhe n e t ofi p o i n d 0 6 0.
d o h each k e 06
0 id and o n d y
46
Phoo6. I f u f i x e s every l i n e o f 0 then, by [24],3.1, a c e n t r a l i z e s T ( k ) f o r each
~ € 0 Conversely, . assume n c e n t r a l i z e s T(h) f o r each J L C O and t h e r e e x i s t s 6 E 0 such t h a t a ( n ) # n . T(o) f i x e s every p o i n t o f n and every l i n e p a r a l l e l t o 4 . L i k e wise, T(n) = a - ’ T ( n ) a f i x e s e v e r y p o i n t o f u ( n ) and every l i n e p a r a l l e l t o u ( n ) . Since n # a ( 4 ) , i t i s easy t o see t h a t t h i s y i e l d s T(n) = <1>, which i s impossible. Now l e t a. be an automorphism o f 0 f i x i n g every l i n e o f 0 and assume a(P) = P f o r
some p o i n t P. I f
JL
i s a l i n e through P and A / / & ,
6 E 0 then u ( b ) =
n and so a ( & ) / / &
which i m p l i e s ~ ( h =) fi. Therefore, a f i x e s every l i n e through P. L e t Q be a p o i n t d i s t i n c t from P and assume PQ=q{ 0. I f w denotes the l i n e o f 0 through Q, we have t h a t a ( Q ) = a ( q f \ w ) = a ( q ) n u ( w ) = q i ? w = Q. I f , on t h e c o n t r a r y , q t 0 then t h e r e l a t i o n a ( Q ) = Q can be obtained by u s i n g t h e same argument as above by s t a r t i n g from a p o i n t P ’ { 4 . The t h e s i s a = I now f o l l o w s . Now we s h a l l g i v e some examples o f Lingenberg S - p a r t i t i o n s . We assume the reader i s acquainted w i t h t h e s t r u c t u r e o f groups SL(2,q); PSU(3,qz), q = ph p a prime; S Z ( ~ ~ ~ ’ R ’ )( 3; 2 n f ’ ) , n > l , and a l s o w i t h the elementary
,
p r o p e r t i e s o f l i n e a r groups. General references are i n [ll] and [12]. I n p a r t i c u l a r , f o r Suzuki groups S Z ( ~ ‘ ~ ” ) , Ree groups R(3“”)
and PSU(3,q’)
see [20], [22] and
[23], [7]r e s p e c t i v e l y . EXAMPLE I. G
2
SL(2,q),
q = ph , q>2. L e t P c S (G) and assume T = S = P; then i t i s P
an easy e x e r c i s e t o show t h a t (G,S,T) I and t h a t [G,S,T],
determines a Lingenberg S - p a r t i t i o n o f type
the completion o f [G,S,T],
i s t h e a f f i n e plane over GF(q).
This i s t h e c l a s s i c a l example which i n s p i r e d t h e work o f Zappa [24]. I t a l s o exp l a i n s the d e f i n i t i o n o f t h e p a r a l l e l i s m i n [G,S,T] EXAMPLE 11. G
7
as given i n s e c t i o n 1.
S z ( q ) , q = Z Z n f ’ , ~ 2 1 .L e t P E S2(G) and l e t Z(P) be t h e c e n t r e o f P.
I f we assume T = Z(P) and S = P then (G,S,T)
determines a Lingenberg S - p a r t i t i o n
o f type I.Indeed, as i t i s w e l l known, NG(T) = N ( P ) and i f x { N G ( T ) then G NG(T)nTX =
M . Biliotti
74 g =
h,.t:
= h2RY w i t h b 1 , n 2 e S, R l , . t t 2 E T and hence
n;'n,
=
d1.t;1 1 x .
If. t , f l , t n f l
and G i s regarded as a c t i n g i n i t s usual doubly t r a n s i t i v e r e p r e s e n t a t i o n o f degree
q z + l then the element .t$(.t;'JX, being t h e product o f two i n v o l u t i o n s w i t h o u t common f i x e d p o i n t s , f i x e s an even number o f p o i n t s . But h;'o1
l i e s i n a Sylow 2-subgroup
o f G and hence i t f i x e s e x a c t l y one p o i n t which i s a c o n t r a d i c t i o n . As we have p r e v i o u s l y shown, we cannot have R =1 f o r only one 4=1,2 and so R =1 for 4=1,2 and 4
~ E S This . y i e l d s S T x n S T Y = S. We =
S t i l l
have I T 1 = q , IS1 = q ' , :NG(T)I
q 2 ( q - I ) , I G / = ( q z + l ) q 2 ( q - I ) and hence, i f T X ' ,
...,TXQ2
=
are t h e q 2 subgroups o f
G which are conjugate t o T and d i s t i n c t f r o m them, i t i s e a s i l y seen t h a t t h e f o l -
lowing r e l a t i o n holds:
4'
1 ( I S T ' ~- ~ IS^)
This proves the a s s e r t i o n .
t
i- I
The completion o f the A-structure [G,S,T]
I N ~ ( T )=I I G ~ .
i s a r e s o l v a b l e S t e i n e r system w i t h
parameters (q(q2-q+I ) , q ) . EYAMPLE 111. G = P X
2
PSU(3,qz), q = 2 h , h > l . L e t Pe S2(G). I t i s w e l l known t h a t NG(P) =
C, where C il c y c l i c o f order ( q 2 - I ) / d with d = ( 3 , q + l ) . Denote by Cl t h e sub( q t i ) / d and s e t T = Z(P), S = P X C,. Then (G,S,T) determines
group o f C of order
a Lingenberg S - p a r t i t i o n o f type I.Indeed, we have again NG(T) = NG(P) and, i f
x f NG(T), N G ( T ) n T X = <1>, so t h a t NG(T)nSTX = 5. Now l e t TX, Ty be such t h a t T#TX+TYgT. By w e l l known p r o p e r t i e s o f G, we have t h a t M =
I. (M,T,T) determines a Lingenberg S - p a r t i t i o n o f type I and
have seen i n Example
hence T X T Y n T = c l > . I t f o l l o w s t h a t T X T Y n S = <1>and t h e r e f o r e S T X n S T Y = S. The t h e s i s can now be achieved by a c a l c u l a t i o n s i m i l a r t o t h a t c a r r i e d out i n Examp l e 11. i s a r e s o l v a b l e S t e i n e r system w i t h
The completion o f the A-structure [G,S,T]
I n t h e case q=Zh, with h even, t h i s S t e i n e r system has
parameters ( q ( q 3 - q 2 + I ) , q ) .
been already obtained by Schulz [la]. EXAMPLE I V . G
2
R(q), q=
32n+I
,
~ 2 1 .We s h a l l make use o f t h e r e p r e s e n t a t i o n o f G
i n P G ( 6 , q ) due t o T i t s [22],§5. L e t xI,x2,. ..,x7 be a coordinate system f o r P G ( 6 , q ) 3n+I Furthermore, l e t I be t h e hyperplane o f x + x
and l e t o c A u t ( G F ( q ) ) , a :
.
P G ( 6 , q ) o f equation x , = O and denote by A the a f f i n e space obtained from P G ( 6 , q ) by assuming
I as the i d e a l hyperplane. Then x=x,/x7,
y=x2/x7,
z=x3/x,,
u=xs/x7,
u=xs/x7,
w=xb/x7
i s a non-homogeneous coordinate system for A . F i n a l l y , s e t and denote by
r -
( m ) : ( I ,O,O,O,O,O,O) { ( m ) } the s e t o f p o i n t s o f A whose coordinates s a t i s f y t h e
equations
(1)
- X Z t yo p y a - za f xy' xzo - x o + i y
u = xzy
p + 3
u
t
=
1o =
yz -
X'CJ'3 x2y2
-
- zz
X ~ a f 4
75
S-Partitions of Groups and Steiner Systems Then G
PGL(6,q)r
2
and G a c t s on
r
i n i t s usual doubly t r a n s i t i v e r e p r e s e n t a t i o n
o f degree q 3 + l . L e t P be t h e unique Sylow 3-subgroup o f G l y i n g i n G(,+.
By u s i n g
t h e r e s u l t s o f T i t s [ 2 2 ] , § 5 , about t h e r e p r e s e n t a t i o n o f t h e elements o f P as w e l l as the f a c t t h a t I Z ( P ) I = q , i t i s n o t hard t o prove t h a t t h e p r o j e c t i v i t i e s l y i n g i n Z(P) are e x a c t l y those o f t h e form t c : (XI,xZ,x3,X4,x-,xb,X-)
+
+ ( X I, X L , X 3 + C X 7 , - C X I+ & + , c x ~ + X ~ - C ' X ~ ,C'X1-2CX3+X6-CZX7
c
,X7),
E
GF(r().
According t o T i t s [ 2 2 ] , § 5 , we have a l s o t h a t
: ,xZ,xJ,X4,x5,X6,x7) (X5,xb,X3,x2,xl ?-x7,-X6) i s an i n v o l u t o r i a l p r o j e c t i v i t y o f G which does n o t l i e i n G
fA
(a).
Therefore,
uZ(P)W i s the c e n t r e o f a Sylow 3-subgroup Q o f G which i s d i s t i n c t from P. Now
i t i s our aim t o prove t h a t i f c , d c GF(q), c + O , d#O, then
'""tCi does dd
n o t belong
t o any Sylow 3-subgroup o f G. Since NG(P)nNG(Q) = E, where E i s c y c l i c o f order
q-l
, and
Z(P)E i s a Frobenius group w i t h Frobenius k e r n e l Z(P) (see [23],111.4),
we can suppose, w i t h o u t
loss o f g e n e r a l i t y ,
d=l. We then have
dCdI : (X,,X2,X3,X,,Xj,X6,XI) ( X I + CX, + coxE,x 2 - cx j, ( J + ZC) x g X-'C - ( C + c P ) x 6 + x ,, - x + ( I - c)x, -c"xE,x 2 - 2cx3+( I -C+C~)X,+C'X 6-X7, X,-(2*2C)X3+CX,,'CuXSf
( It2CtC"c')X6-X7
2CX3'CoXg-c2Xgfx7)
A s t r a i g h t f o r w a r d c a l c u l a t i o n shows t h a t d c d Jpossesses the eigenvalue I whose
eigenspace i s generated by t h e v e c t o r ( 0 , I ,l/'Z,-cu~*,O,l/c,l).
Now suppose d c d J
l i e s i n a Sylow 3-subgroup o f G, then t h e f o l l o w i n g hold: - ili.tc does dI n o t have any eigenvalue d i f f e r e n t from I , f o r we are i n charact e r i s t i c 3;
-
LK
C
wx
I
must f i x a p o i n t o f
r
- {
(m)}.
From t h a t which we have proved p r e v i o u s l y , we can i n f e r t h a t t h e f i x e d p o i n t o f
d2dl
on
r -
{ (a)] must have non-homogeneous coordinates ( O , l , I /2, - c O - ~ ,0, I /c)
.
But these coordinates do n o t s a t i s f y (l), a c o n t r a d i c t i o n . Now we may argue as i n the previous examples t o show t h a t i f we s e t T = Z ( P ) and S = P then (G,S,T) determines a Lingenberg 5 - p a r t i t i o n o f type The completion o f t h e A-structure parameters ( q ( q 3 - q '+ I ) ,q 1. EXAMPLE V. G = SL(n,q),
[G,S,T]
I.
i s a r e s o l v a b l e S t e i n e r system w i t h
q = ph , n>3. L e t K = GF(q), V = K" and U a 1-dimensional
subspace o f V. Denote by T ( g , p ) t h e t r a n s v e c t i o n ptHomK(V,K) w i t h p(2) = T(U) =
0,p#O.
y
-f
l-u(l)awhere g c
b of u(b)=gj.
For a f i x e d non-zero v e c t o r
{I, T(b,u) : O # U E HomK(V,K),
V and
U, s e t
Then T(U) i s a subgroup o f G (see [11],11, H i l f s s a t z 6.5). F i n a l l y , denote by S(U) t h e subgroup o f G f i x i n g U pointwise. Then (G,S(U),T(U))
determines a Lingenberg
S - p a r t i t i o n o f type 11. I t i s indeed enough t o observe t h a t t h e A - s t r u c t u r e which i s obtained from t h e a f f i n e space A associated t o V by removing t h e o r i g i n 0
i s a q u a s i - t r a n s l a t i o n A - s t r u c t u r e w i t h respect t o the p e n c i l 0 o f the l i n e s
M.Biliotti
16 through
0 (disregarding
the point
which a r e conjugated t o T(U).
0)and
t o t h e f a m i l y 3 o f t h e subgroups o f G
Furthermore, a t r a n s v e c t i o n o f T(U) with hyperplane
ff f i x e s a l l the l i n e s o f 0 l y i n g i n
H , so t h a t
T(U) i s n o t semiregular on 8 . The
a s s e r t i o n now f o l l o w s from P r o p o s i t i o n s 1.2 and 2.2.
3. FURTHER RESULTS ON LINGENBERG S-PARTITIONS OF TYPE I We w i l l r e q u i r e t h e f o l l o w i n g lemma. LEMMA 3.1.
Let G be one 06 t h e 6oCCowing ghoupn: h
SL(?,q), q = p , p p’Lime, 4 2 4 ; S z ( q ) , q = p2 n f 1 p=2, @ I ; S U O , ~ ~,) q - p
i,, h
PSU(~,~’), q=p
p phime, 4 > 2 , 3 1 ~ 7 ;
, P phime,
~22;
R(q), q=pZn*l, p=3, el; and l e i P be a SgCow p-nubgmup
conditioMn: (I1 /TI - I (21
TnZ(G)
0 4 G. 16
T
a nomat dubgtoup
06
NG(P) b ~ ~ q 4 u 2 g
[NG(P):Tl and =
then T = Z(P).
Pmod.
We s h a l l i n v e s t i g a t e t h e v a r i o u s cases separately.
L e t G = sL(z,q),
q s 4 . Assume q i s odd. Then I Z ( G ) ( = 2, ] P I = q , ING(P)I = q ( q - 1 )
= N (P)/Z(G) i s a Frobenius group w i t h Frobenius k e r n e l p. Since T Q N, then G Satz 8.16, we have t h a t e i t h e r 7 < or h P. I n t h e f i r s t case, i t
and
r
by [ l l ] , V ,
f o l l o w s t h a t T < PZ(G). But, T n Z ( G ) = <1>and hence T < P, which i m p l i e s T =
/ N G ( P ) / = q 2 ( q - l ) , lP1 = q z ,
I Z ( P ) I = q . We have t h a t e i t h e r T 2 P or T c P. Con-
d i t i o n ( 1 ) i s u n s a t i s f i e d when T 2 P. I f T < P, then e i t h e r T 2 Z(P) or Tn Z(P) =
1
= <1>s i n c e Z(P) i s a minimal normal subgroup o f NG(P). As T Q NG(P), we have t h a t
q-l
IT[-I.
So, i n t h e former
case, i t f o l l o w s T = Z(P) from c o n d i t i o n (1). I n
the l a t t e r case we have P = T X Z(P). However, P/Z(P) i s a b e l i a n and hence P must be a b e l i a n which
is a c o n t r a d i c t i o n .
L e t G = S U ( 3 , q 2 ) , 3 1 q + 7 . We have /Z(G)I = 3 and N (P) = P X C , where / P I = 9’ C i s c y c l i c o f order q2-I and c o n t a i n s Z(G).
-
G
Moreover, I Z ( P ) )
N = NG(P)/Z(P)Z(G) i s a Frobenius group with Frobenius k e r n e l plements isomorphic t o clearly
p
c. Since io N, we must have e i t h e r
i s a minimal normal subgroup o f
and hence
7=
and
= q.
and Frobenius com<
p or
r 2 P. B u t ,
<1> i n t h e f i r s t case.
By c o n d i t i o n ( 2 ) and since ( 3 , q ) = I , we then have t h a t T 5 Z ( P ) and thence T = Z(P)
77
S-Partitions of Groups and Steiner Systems because Z(P) i s a minimal normal subgroup o f NG(P). I n t h e l a t t e r case we cannot have T 2 P by c o n d i t i o n
(l), w h i l e T n Z ( P ) = <1> forces P t o be ( T n P ) Z ( P ) , b u t
as we have seen b e f o r e then P must be a b e l i a n which cannot be t h e case. L e t G = PSU(3,q').
The p r o o f i s s i m i l a r t o t h e previous one.
Let G = R(q).
We have NG(P) = P X C, where ( P I = q 3 and C i s c y c l i c of order q - 1 . = NG(P)/P' i s a FrobenMoreover, Z(P) < P' = @ ( P I , IZ(P)I = q , I P ' I = q 2 . Since i u s group w i t h Frobenius k e r n e l have e i t h e r
7
=
<1> or
7 2 p.
p
(see [23],111.11),
as i n t h e p r e v i o u s cases, we
I n t h e f i r s t case T 5 P I . I f T n Z ( P ) #
T 2 Z(P) since Z(P) i s a minimal normal subgroup o f NG(P), b u t by [ 2 3 ] , I I I . 2 , T > Z(P) i m p l i e s T = P ' and c o n d i t i o n (1) i s n o t s a t i s f i e d . So T = Z(P). We can-
n o t have TfiZ(P) = -1> s i n c e if X E PI der 2 q z (see [23],111.2)
-
Z(P) then i t s c e n t r a l i z e r i n NG(P) has or-
and hence I T ( > q , c o n t r a r y t o T S P I . I n t h e l a t t e r ca-
se, we cannot have T n P ' = <1>, since for each t Z(P) < PI (see [23],
[23],111.2)
xE
P - P' we have o ( x ) = 9 and
Theorem). Nevertheless, T n P ' # <1> i m p l i e s I T 1 2 q'
x3 6 (see
and again c o n d i t i o n (1) i s n o t s a t i s f i e d . This completes the p r o o f .
The f o l l o w i n g theorem i s concerned w i t h Lingenberg S - p a r t i t i o n s o f type I i n t h e case o f T being o f even order. THEOREM 3.2.
L e L (G,S,T)
d e t w i n e a ling en be^ S-pcmLiAon
06
even o t d m t h e n one 0 6 t h e doLLowing h u t & : G = O(G)T und T iA a Fhobeniun cornpLement; (u) h ( 6 . 1 ) G 2 SL(2,q), 9.2 , h62; T = S = P w i t h P t S2(G); ( b . 2 ) G 2 SZ(~),q - 2 2ntJ a21; T = Z(P), S = P w i t h P6S2(G); i, ( 6 . 3 ) G 2 PSU(3,qz), q = 2 , h22; T = Z(P) w i t h . P G S2(G) and S = ( q + l l / d , &eke d=13,4+11.
type I.1 6 T ha^
= P X
CI with
IC1I =
Phood. I n t h e A-structure [G,S,T], the p e n c i l o f l i n e s 0 i s a t r a n s i t i v e k s e t 4 = N G ( T ) t 0 then E = Ne(?) and, by P r o p o s i t i o n 2.2, ? a c t s semiJl = r e g u l a r l y on 0 - { a } . Then by [ l o ] , Theorem 2, e i t h e r t h e case ( a ) occurs or 2 S L ( ~ , C ( ) , S Z ( C ( ) , PSU(3,qz), SU(3,q') w i t h q = Z h , h > l . I n the l a t t e r case by [ l o ] , Lemma 3, E a c t s on 0 i n i t s u s u a l doubly t r a n s i t i v e r e p r e s e n t a t i o n o f degree q + l , q 2 + 1 , q 3 t 1 , q 3 + 1 r e s p e c t i v e l y . Then, i t is w e l l known t h a t h = Nc(F) with P"eS2(G) and hence Nc(?) = N c ( B ) . By t a k i n g account o f P r o p o s i t i o n 2.2, we see t h a t 7 s a t i s f i e s c o n d i t i o n s (1) and ( 2 ) o f Lemma 3.1. Therefore ? = Z ( P ) . When E = SL(Z,q), w i t h 10l>l. I f
e
Sz(q)
E
or
PSU(3,qz) we o b t a i n (b.1) - (b.3) i n view o f P r o p o s i t i o n 2.2,(3).
= SU(3,q2) w i t h 31qtJ we have
171
= IZ(p)l =
If
q and hence I ~ l ( = q - l Since . 1Z(G)I=3,
t h i s i m p l i e s t h a t 3 Iq- I by P r o p o s i t i o n 2.4, a c o n t r a d i c t i o n . Therefore, t h e case
G"
= SU(3,q2), w i t h 31q+l, cannot occur.
We p o i n t o u t t h a t Examples
I, I 1 and I11 o f s e c t i o n
2 show t h a t t h e cases (b.11,
(b.2) and (b.3) a c t u a l l y occur. On the c o n t r a r y , i t seems very d i f f i c u l t t o achieve a complete c l a s s i f i c a t i o n o f Lingenberg S - p a r t i t i o n s o f type I i n t h e case ( a ) .
M. Biliotti
78
I n succession, we g i v e some r e s u l t s and examples concerning t h i s case.
LeA (G,S,T)
PROPOSITION 3.3.
d&tehmine u Lingenbehg S - p a d t i o n ud t y p e I und
M-
E induced
a F h o b e u p m u t a t i o n gmup on t h e pen& 0 i n %he A-n&ucRuhe [G,S,Tl, then Rhe ~oUoiu4ngh o l d : h ( I )G = M X T, whehe M A a nonabe14un n p e c b l p-ghoup 0 4 m d u q 2 m t ’ w i t h q = p ,
dume [ T I 2 3.
16
m,hLJ; ( 2 1 IZ(G)I = lZ(M)I = 4,
I T 1 = q + I , S = T,
NG(T) = TZ(M).
P m a 6 . By P r o p o s i t i o n 2.4, Z(E) i s t h e k e r n e l o f t h e r e p r e s e n t a t i o n o f E on 0. Go = E / Z ( c ) and a c t s on 0 as a Frobenius group by our assumptions.
c’
Therefore, Denote by
G
the Frobenius k e r n e l o f
By P r o p o s i t i o n s 2.2 and 2.4,
IT], I i l ) = I
Moreover,
since
= G/Z(G) and l e t M
we have t h a t IZ(G)I
I /TI-1
i G such t h a t M/Z(G) =
M.
and hence ( l T l , l Z ( G ) l ) = l .
i s contained i n a Frobenius complement o f
c.
There-
<1> and MT = M X T. I f x t G then, c l e a r l y , T X C M T and hence G = MT and F = i@ . We have = G 2 NG(T) 2 T, so t h a t T = NG(T) and NG(T) = TZ(G). Set (TI = R , then ( Z ( G ) ( = [NG(T):T] 2 [NG(T):S] = t-I by P r o p o s i t i o n 2.2. Since, fore, T n M =
r
as we have p r e v i o u s l y seen, IZ(G)I
ii
Note t h a t since
group of M with P
I
R - 1 , i t f o l l o w s t h a t / Z ( G ) I d-I and S = T .
i s n i l p o t e n t so i s M (see [11], V.a.7).
$
Z(G) and l e t N = PT. Since (G,S,T)
L e t P be a Sylow p-sub-
determines a Lingenberg
S-partitionof type I, t h e f o l l o w i n g r e l a t i o n holds
It‘
(2)
- $1
!nlRc - I )
+
Rc
c
y1
,
= I N ( and c = ( P n Z ( G ) ( . From (2), i t f o l l o w s t h a t t - l j c since n>tc and
where
hence c=X-I and Z(G) < P. This y i e l d s M = P. Consider the commutators o f t h e form [x,g]
with X E T and g t M-Z(G).
We have [x,g]
and hence [ X , g ]
= X-’(g-’xg)
E
TT’.
Each complex TTg c o n t a i n s e x a c t l y t-J n o n - i d e n t i c a l d i s t i n c t commutators o f t h e form [ x , g ] .
Moreover, i f T 9 # T6 then T T g n T T 6 = T and hence, t h e R-7 commutators
l y i n g i n TTg are d i s t i n c t from those l y i n g i n TT‘. by s e t t i n g [M,T].
(GI
Since
= 6 there e x i s t a t l e a s t ( X - I ) ( i - 7 ) t I
Since [ M , T ]
IITX
: X E G}
2 M and I M I = m ( X - I ) , i t f o l l o w s t h a t [ M , T l
= M.
t h e r e e x i s t s a c h a r a c t e r i s t i c a b e l i a n subgroup A o f M such t h a t A group AT contains e x a c t l y a = [ A : Z ( G ) n A ]
2 (t-I)(a-I)tJ
13.4(b),
121,
then
6
Now suppose Z(G). Then the
,
( t - I ) a . The l a t t e r r e l a b u t t h i s c o n t r a d i c t s a r e s u l t o f Zassenhaus [11],III, Sat2
where a>?. From t h i s i t f o l l o w s t h a t A = [ A , T ] t i o n y i e l d s Z(G) < A ,
=
d i s t i n c t conjugate elements of T. By
using the same argument as before, we have ( t - I ) a 2 / A \ 2 I[A,T]I
I
d i s t i n c t commutators l y i n g i n
and ( A ( =
since IZ(G)) > I. Therefore, a c h a r a c t e r i s t i c a b e l i a n subgroup o f M i s
c e n t r a l i n G. I n conclusion we have proved t h a t :
(I) ( I M I , I T I ) = I , (11) [ M , T l = M, (111)
T c e n t r a l i z e s every c h a r a c t e r i s t i c a b e l i a n subgroup o f M.
By a r e s u l t of Thompson [ L l ] , I I I ,
Satz 13.6, we then have t h a t M i s a nonabelian
79
S-Purtitiorisof G r o i q s arid Steirier Systems
special p-group. Moreover, since Z(M) is a characteristic abelian subgroup of M, phtn. Since a we have that Z(M) = Z(G). Let lZ(G)I = t - 1 = p h , h21, and let l M l Frobenius complement of G/Z(G) has order . t = p h + I , it follows that p h +llp"-l. From so this, we have that ph+ I I pn+ph=ph (pn-h+1 ) and hence ph+ 1 I pn-h- Ph =Ph (Pn-2h- I ph+ 1 1 p n - 2 h - I . Let b E P such that bhSn< Ib+ lih. By iterating the above procedure, it . completes the proof. is easy to prove that b must be even and P ' - ~ ~ - I = O This
.
A Lingenberg S-partition of type I satisfying conditions ( 1 ) and (2) of Proposition 3.3 and its associated A-structure will be called 4peciu.e. An example is given below. EXAMPLE VI. Ue assume the reader is familiar with [14],V,§32. Let ~1 be the projective plane over GF(qz), q=ph , and let p be a hermitian polarity of TI. It is well known that the absolute points and non-absolute lines of p make a Stciner system u with parameters b=q+l and v = q 3 + 1 , which is usually called the d u b b i c d unitul. Moreover the group P(U) consisting o f the projectivities of TI leaving U invariant is isomorphic to PGU(3,qZ). According to Bose [ 5 ] , § 6 , for each absolute line p of [ I , we may define a parallelism among the lines of U as follows: a class of parallel lines consists of a non-absolute line fi through p ( p ) and the non-absolute lines through p ( h ) . Note that p ( p ) E h implies p ( 8 ) 6 p. Therefore, the group T ( h ) consisting o f all ( p ( h ) ,&)-homologies lying in P(U) preserves the parallelism just defined in U , because it fixes the line p . The group T(a) fixes each line i? parallel to h and acts regularly on the points o f L lying in U because T(h) has order 9 t J . Moreover, there exists a unique Sylow p-subgroup M of P ( U ) which fixes p ( p ) and so p itself and acts transitively on the Q' non-absolute lines through p ( p ) . It follows that U - {PI is a quasi-translation A-structure with respect to the pencil 0 of non-absolute lines through p ( p ) and to the family 8 = { T ( h ) : h E O}. It is easily seen that: - G =
= M X T(b),
- T(A) acts semiregularly on o
b E @
;
- [A} ;
- IZ(M)I= q and Z ( M ) consists of all (p(p),p)-elations lying in P(U) and therefore
it fixes every line of 0; scts regularly on 0. From this, it follows that Z ( M ) = Z(G) and G/Z(G) is a Frobenius group. Therefore, (G,T(h),T(h)) determines a special Lingenberg S-partition. - I M I = q 3 and M/Z(M)
now consider case (a) of Theorem 3.2. Assume, hebpecR 20 .the bubghoup T 141 Z 0 u L i ~ g e ~ b e hSg- p ~ ~ h L i . t i 0o6~ G 0 6 .type I ~ i R h w i t h IT I t 3 and (44)G = CT, whefie C 0 bo.&ubi?e and C Q G.
We
be the pencil of lines of [G,S,T]. By Propositions 2.2 and 2.4, we have that and hence Z(c) < ?. Let c/Z(E) be a minimal normal subgroup of G/Z(G) contained in e/Z(c). Since $ Z ( E ) and E/Z(c) acts transitively on 0, then, Let 0
(/TI,JZ(c))) = 1
e
M . Biliotti
80
i s the o r b i t o f
i f 4 = NG(T)C 0 and since
k(E)
h under
i s elementary a b e l i a n f o r
t a r y abelian. I t f o l l o w s now t h a t
?
r,
we have t h a t Is11 > 1 . Moreover,
i s solvable, i t f o l l o w s t h a t
a c t s r e g u l a r l y on
R i n v a r i a n t and a c t s semiregularly on s1 -
LR
R. Moreover, ?R =
[ k } . From t h i s , we i n f e r t h a t
i s elemen-
7
leaves
?ITn
is a
Frobenius group. Now s e t F =
IN,(
6.
Since C i s a Lingenberg 5 - p a r t i t i o n o f G o f type I,
the f o l l o w i n g r e l a t i o n holds: (3)
(,t~o
S
- I ) t no 2 6. t-1. On t h e o t h e r hand we have t h a t
- bo)(d/na
From t h i s , i t f o l l o w s t h a t n o / b o t
S
Therefore, n o / b o = t-f and ( 3 ) h o l d s as an e q u a l i t y . Using t h i s ,
[N:S] = . t - 1 .
i t i s n o t d i f f i c u l t t o see t h a t ifwe s e t R = : x t F 1 } u {NFl(T1)
then C I = { T f
[N,:S,]
respect t o the subgroup TI.
But
GFs;,
s l =SJR,
T ~ IRA, =
F,= F/R,
i s a Lingenberg S - p a r t i t i o n o f F 1 o f type I w i t h
-R
-R-R
F I = I- T
and hence, by P r o p o s i t i o n 3.3,
XI i s a
b p e c i d lingenbekg S - p a t L i L i u n . From a geometrical p o i n t o f view, t h e previous r e s u l t can be expressed as f o l l o w s .
PROPOSITION 3.4.
L e t C be. a Lingenbehg S-pa/ttLtitian a6 G
0 6 type I
w c t h k e ~ p e c tt o
-the hubghaup -i w i t h T 2 3. Adbume G = CT, whem C i h a hoLwabLe n u m d hubghoup
cuntaim a bubbpace w h i c h i~ a b p e c i d
ud G. Then t h e A-bRhuctwle [G,S,T]
A-btkuc-
tuke.
Pkoo6. L e t [FI,S1,T1]
be the s p e c i a l A-structure r e l a t e d t o the s p e c i a l S , - p a r t i -
t i o n & o f Fi described above. I f S I X i s a p o i n t o f [F,,Sl,T1] the map from t h e s e t o f p o i n t s o f [FL,SI,T1] defined as f o l l o w s
n
-
: SIX
-t
sx
and
x = Kx,
l e t q be
i n t o t h e s e t o f p o i n t s o f [G,S,T]
.
I t i s s t r a i g h t f o r w a r d t o show t h a t 11 i s w e l l d e f i n e d and g i v e s an embedding o f [FI,Sl,TIl
i n t o [G,S,Tl.
4 . LINGENBERG 5-PARTITIONS OF DOUBLY TRANSITIVE PERMUTATION GROUPS determines a Lingenberg S - p a r t i t i o n . Since i n Examples I - V vie have
Assume ( G , I , T ) that: (a)
t h r gnoup
cicib
Z-tcanoiLivek?y un t h e pen&
06 f i n e d
0
06
.the A - ~ i h u c . t u c e
[G,S,Tl. then the n a t u r a l question a r i s e s whether i t i s p o s s i b l e t o c l a s s i f y a l l the t I i p l e s
(G,S,T) which determine Lingenberg S - p a r t i t i o n s s a t i s f y i n g c o n d i t i o n ( a ) . I n the f o l l o w i i i g , we s n a l l prove t h a t a r a t h e r s a t i s f a c t o r y answer t o t h i s question may be g i v e n provided t h a t t h e c l a s s i f i c a t i o n o f doubly t r a n s i t i v e permutation groups i s assumed.
As i t i s w e l l knowri, such a c l a s s i f i c a t i o n f o l l o w s f r o m t h a t o f F i n i t e
simple groups.
(G,S,l) deXehmAneb a Lingenbehg S-pamLCi.on C &Lion ( u l . 16 C 0 ad t y p e It h e n une 06 t h e ,joXCawing h d h : THEOREM 4.1.
Annwrie
b a t ~ A 6 y ~ ncung
81
S-Partitionsof Groups and Steiner Systems
(I]
q = p c l , ph22; T = S = P l ~ i h i hP E S (C); P 2n+ I ntl; T = Z(P), S = P wi2h P 6 S 2 ( G ) ; G 2 Sz(q), Q = G 2 PSU(3,qz), q=2', h t 2 ; T = Z(P) wiRh PE S2(G), S = P A C I , wh&he I C I = ( q + l ) / d ,d=(3,y+ll; R ( q ) , q = 3 2 n C 7 , n 2 i ; T = Z(P), S = P w a h PES3(G). G 0 0 6 t y p e I1 lhen PSL(n,q) S G/Z(G) S PTL(n,q) iyhehe n23. SL(Z,q),
G
(21 (31
(4) 18 C
a LinyenbErg 5 - p a r t i t i o n o f type - determines s a t i s f i e s the following condition:
I. By P r o p o s i t i o n
Pk006. Assume (G,S,T) -,@
t l i e group G
2.7,
* G/Z(E)
(h) f o r each h e @ , t h e s t a b i l i z e r o f s e m i r e g u l a r l y on
o-
5
6'
in
I=
c o n t a i n s a normal subgroup which a c t s
{t}.
From t h e c l a s s i f i c a t i o n theorem o f f i n i t e doubly t r a n s i t i v e permutation groups, we
(see [ 4 ] ) i s a c t u a l l y a theorem (see
h a ? t h a t t h e so c a l l e d "Hering conjecture"
[19],p.302)
c0
asserting that i f
i s 2 - t r a n s i t i v e on 0 and s a t i s f i e s ( h ) then one
o f t h e f o l l o w i n g holds:
c0
(j) c o n t a i n s a r e g u l a r normal subgroup, (jj) Eo PSL(Z,(i), q L 4 , Sz(q), PSU(3,q"), q > 2 , or R ( y ) , 4'3, -1
aiid
Go
a c t s on 0 i n
i t s usual doubly t r a n s i t i v e r e p r e s e n t a t i o n . We s h a l l i n v e s t i g a t e these cases separately.
( j ) . Let
C~c.14
hence
Go =
E0
be t h e r e g u l a r rlorrnal subgroup o f
cO. We have t h a t Go
" G Z ( e ) i s a Frobenius group. If( T I = 2 then
then by P r o p o s i t i o n 3.3,
-,o
we must have I N
I
=
EB =
and
=
= SL<2,2). I f ( T I 1 3
q Z m , where q i s a prime power and m 2 1
tdureoveL, ( T I = q + ~ Since . Go i s 2 - t r a n s i t i v e on
B, i t
iollows that q2m-l=q+l.
T h i s i m p l i e s q = 2 , m = l and i t i s very easy t o show t h a t G = SL(2,3). Cane ( j j ) . Note t h a t i f L 5 G and LZ(G) = G then L = G s i n c e ( \ T l , l Z ( G ) l ) = I and G i s generated by T and i t s conjugates. Therefore, G i s a c e n t r a l i r r e d u c i b l e
tension o f
$).
[11],[12]) then t h e r e e x i s t s a unique r e p r e s e n t a t i o n group H o f of Z(H). Moreover, Z(G) = Z(H)/Z,. I f M(E')
f o r some subgroup Z, multiplier o f 2.2,
ex-
Since, i n t h e case under c o n s i d e r a t i o n , G i s a simple group ( s e e
EB then
I
IZ(H)I = IM(CO)I ( s e e [21],Ch.2,59).
we have t h a t ( Z ( G ) I
assume :
IT1
- I,
i t f o l l o w s t h a t IZ(G)I
I
? and
G = H/Z,
denotes t h e Schur
Since, by P r o p o s i t i o n ( l M ( ~ O ) ~ , ~ T l - JNow ).
q = ph , h 2 2 . I f p i s odd then we cannot have G PSL(2,q) s h c e i n t h i s case every normal subgroup o f NG(P), where P e S (C), c o n t a i n s P and hence r e P l a t i o n ( 3 ) o f P r o p o s i t i o n 2.2 cannot be s a t i s f i e d f o r \ P I = q and ING(P)I = % q [ q - I I . Therefore, when q:J,9 by [11],V,25.7, we must have G = H = SL(2,q) and t h e t n e s i s f o l l o w s from Lemma 3.1. If9 - 4 then T has even order and, by Theorem 3.2, G = = S L ( 2 , 4 ) . L e t q - 9 . Since NG(T)/Z(G) is a Frobenius group o f order 9.4 with Fro-
'"G
= PSL(p,(i),
benius k e r n e l o f order 9 , i t f o l l o w s t h a t I T 1 I 9 . On t h e o t h e r hand,
-
where
101= I0 and hence IT
fore,
/ Z ( G ) I = 2 and G
I
= 9 . By [8],Table 1, we have t h a t M(Co)
SL(2,9).
IT1
= Z6.
I
I01-J There-
82
M . Riliotti
Let
Go --
Sz(g). I n t h i s case T has even order and (2) f o l l o w s from Theorem 3.2.
to
= PSU(3,q2), q > 2 . By [ 9 ] , Theorem 2, we have e i t h e r G 2 PSU(3,q') o r G 2 SU(3,q2), 3 1 q + l , may be excluded by arguing as i n the = SU(3,q'). The case G h p r o o f o f Theorem 3.2. Assume G ^1 PSU(3,qZ), y = p , p an odd prime. By Lemma 3.1, we Let
have t h a t T = Z(P) with P t S (G) and S = P X
P
C1,
where C I i s c y c l i c of order
g+J/d,
d=(3,q+J). Moreover C , < C, where C i s c y c l i c o f order q2-J/d and C < NG(P). L e t C2 be t h e subgroup o f C of order q - J . As i t i s w e l l known, t h e r e e x i s t s a subgroup M o f G such t h a t MnNG(P) = TC2 and SL(2,q)
= M =
Let b = = .ti, where I f - t e T and i is t h e unique i n v o l u t i o n i n C2. As was shown i n Example I,
subgroup Tx o f T, where TX may be chosen i n such a way t h a t Cz < NM(Tx). determines a Lingenberg S - p a r t i t i o n o f type
(M,TX,TX)
we have t h a t b
b
E
{ NW(TX). Therefore, t h e r e e x i s t s
I.
Since NM(T)nNM(TX) = C 2 ,
Ty < M with T#TY#Tx such t h a t
TXTY and hence T x T y n S # < l > since b E S. Note t h a t
i E Ci
f o r q odd. I t f o l l o w s
does n o t determine a Lingenberg S - p a r t i t i o n . The t h e s i s can now be
t h a t (G,S,T)
achieved by using Lemma 3.1. Let
Go
R ( g ) , q>3. By [l], Theorem 1, we have t h a t M(?)
= clz and ( 4 ) again f o l -
lows from Lemma 3.1. Now assume (G,S,T)
--
determines a Lingenberg S - p a r t i t i o n o f t y p e I I . I f NG(T)x i s a
l i n e o f 0 then the s t a b i l i z e r NE(T"X)' o f t h i s l i n e i n group Tx'
Eo
c o n t a i n s the normal sub-
s a t i s f y i n g the c o n d i t i o n
(1) yX0f3?@=
?@such
that
T"yo#T"xxo.
Indeed, assume ( 1 ) does n o t hold. Then TXZ(G)i7TYZ(G) > Z(G) f o r some Ty with TX# fTy.
I t f o l l o w s t h a t T X T Y n Z ( G ) # < l >since TXnTY=
sists z
E
Z(G),
= S then z
E
-1
~ $ 1 ,such t h a t z c T T Y X E STY'-'.
Since z
E
NG(T) and STY'-'
n NG(T)=
5, b u t t h i s i s a c o n t r a d i c t i o n because Sn Z(G) = <1>. Moreover, by
P r o p o s i t i o n 2.2, we have:
(2)
' 3
does n o t a c t semiregularly on Q - {NG(T)x].
By a w e l l known r e s u l t o f O'Nan [ 1 5 ] , PSL(n,q) 5
6 PrL(n,q)
Theorem A,
c o n d i t i o n s (1) and ( 2 ) i m p l y
with n23. So t h e t h e s i s f o l l o w s from P r o p o s i t i o n 2.4.
As a f i n a l remark, we note t h a t t h e c l a s s i f i c a t i o n theorem o f doubly t r a n s i t i v e permutation groups i s r e q u i r e d o n l y when (G,S,T)
determines a Lingenberg S - p a r t i t i o n
o f type I and T has odd order.
REFERENCES.
[l] A l p e r i n , J.L. and Gorenstein, D., The m u l t i p l i c a t o r s o f c e r t a i n simple groups, Proc. Am. Math. SOC. 17 (1966), 515-519. [ 2 ] Andre, J., Uber P a r a l l e l s t r u k t u r e n , T e i l I : Gundbegriffe, Math. Z. 76 (1961),
85-102.
S-Partitions of Groups and Steirier Systems
[ 31
Andr6, J.,
83
Uber P a r a l l e l s t r u k t u r e n , T e i l I 1 : T r a n s l a t i o n s s t r u k t u r e n , Math. Z.
76 (1961), 155-163.
[41 Aschbacher, M., F-sets and permutation groups, J. Algebra 30 (1974), 400-416. [5] Bose, R.C., On t h e a p p l i c a t i o n o f f i n i t e p r o j e c t i v e geometry f o r d e r i v i n g a
c e r t a i n s e r i e s o f balanced Kirkman arrangements, in:The Golden Jub. Comm., C a l c u t t a Math. SOC. (1958-591,341-354. [6] B r e n t i , F., S u l l e S - p a r t i z i o n i d i Sylow i n alcune c l a s s i d i g r u p p i f i n i t i , Boll. Un. Mat. I t . (6) 3-8 (1984), 665-685. [ 71 Burkhardt, R., Uber d i e Zerlegungszahlen der u n i t a r e n Gruppen PSU(3,2 2 f 1 , J. Algebra 61 (19791, 548-581. [ 8 ] Griess, R.L.,Jr., Schur M u l t i p l i e r s o f t h e known f i n i t e simple groups, B u l l . Am. Math. Soc. 78 (19721, 68-71. Schur M u l t i p l i e r s o f f i n i t e simple groups o f L i e type, [9] Griess, R.L.,Jr., Trans. Am. Math. SOC. 183 (1973), 355-421. [ 101 Hering, C., On subgroups w i t h t r i v i a l normalizer i n t e r s e c t i o n , J. Algebra 20
(19721%622-629. [ll] Huppert, B., Endliche Gruppen I(Springer-Verlag, Berlin-Heidelberg-New York, 1979). [ 121 Huppert, B. and Blackburn, N., F i n i t e Groups I11 (Springer-Verlag, B e r l i n -Heidelberg-New York, 1982). [ 131 Lingenberg, R. , Uber Gruppen p r o j e c t i v e r K o l l i n e a t i o n e n , whelche e i n e perspect i v e D u a l i t a t i n v a r i a n t lassen, Arch. Math. 13 (1962), 385-400. [ 141 Luneburg , H., T r a n s l a t i o n Planes (Springer-Verlag, Berlin-Heidelberg-New York, 1980). [ 151 O'Nan, M.E., Normal s t r u c t u r e o f t h e one-point s t a b i l i z e r o f a doubly-trans i t i v e permutation group. I, Trans. Am. Math. SOC. 214 (19751, 1-42. [16] Rosati, L.A., S u l l e S - p a r t i z i o n i n e i g r u p p i non a b e l i a n i d ' o r d i n e pq, Rend. Sem. Mat. Univ. Padova 38 (19671, 108-117. [17] S c a r s e l l i , A., S u l l e S - p a r t i z i o n i r e g o l a r i d i un gruppo f i n i t o , A t t i ACC. Naz. L i n c e i , Rend C1. S c i . F i s . Mat. Nat. ( 8 ; 62 (1977), 300-304. [18] Schulz, R.H., Zur Geometrie der PSU(3,q ) , i n : B e i t r a g e zur Geometr. Algebra, Proc. Symp. Duisburg, 1976 (Birkhauser, Basel, 1977), 293-298. [19] Shult, E.E., Permutation groups with few f i x e d p o i n t s , i n : Geometry - von S t a u d t ' s P o i n t o f View, Proc. NATO Adv. Study I n s t . Bad Windsheim, 1980 ( 0 . R e i d e l P.C., Oordrecht, 19811, 275-311. [20] Suzuki, M., On a c l a s s o f doubly t r a n s i t i v e groups, Ann. Math. 75 (19621, 104-145. [ 211 Suzuki, M., Group Theory I (Springer-Verlag, Berlin-Heidelberg-New York, 1982) [22] T i t s , J., Les groupes simples de Suzuki e t de Ree, i n : Sem. Bourbaki, 13e annCe, 210 (1960/61), 1-18. 1231 ~- Ward, H.N., On Reels s e r i e s o f simple groups, Trans. Am. Math. SOC. 121 (19661, 62-89. 1241 ZaDOa. G.. S u a l i sDazi a e n e r a l i quasi d i t r a s l a z i o n e , Le Matematiche (Catan i a ) i9 (i9647, 127-143: S u l k S - p a r t i z i o n i d i un gruppo f i n i t o , Ann. Mat. Pura Appl. (4) 1251 . . Zappa, G., 74 (19661, 1-14. [26] Zappa, G., P a r t i z i o n i g e n e r a l i z z a t e n e i gruppi, i n : C o l l . I n t . Teorie Comb. 1973 (Acc. Naz. L i n c e i , Roma 1976), 433-437. i
,