iii. the nine dimensional case

MATHEMATICS OF COMPUTATION, VOLUME 34, NUMBER 149

JANUARY 1980, PAGES 245-258

On Maximal Finite Irreducible Subgroups of GL{n, Z)

III. The Nine Dimensional Case

By Wilhelm Plesken and Michael Pohst

Abstract. All maximal finite absolutely irreducible subgroups of GL(9, Z) are deter-

mined up to conjugacy in GL(9, Z).

1. Introduction. We determine all maximal finite irreducible subgroups of

GL(9, Z) up to Z-equivalence. (Here and in the following, irreducible means C-irre-

ducible.)* There are 20 Z-classes, a set of representatives of which is described in Sec-

tion 4, Theorem (4.1). The quadratic forms fixed by these groups are Usted in Sec-

tion 3.

We employ the methods developed in Part I [7]. In Section 2 the minimal irre-

ducible finite subgroups of GZ,(9, Z) are determined up to Q-equivalence; there are

only three classes. The Z-classes of the natural representations of the three groups,

respectively the -<-maximal centerings of the corresponding lattices, were electronically

computed on the CDC Cyber 76 at the Rechenzentrum der Universität zu Köln. They

are listed in Section 3.

In Part V of these series of papers we shall present a full set of representatives

of the Z-classes of the maximal finite irreducible subgroups of G¿(«, Z) for n < 9 by

listing generators of the groups, the corresponding quadratic forms fixed by these

groups, and the shortest vectors of these forms.

2. The Minimal Irreducible Finite Subgroups of CL(9, Z). The minimal irre-

ducible finite subgroups of C7Z,(9, Z) which are solvable will turn out to be rationally

equivalent to a monomial group. Therefore (see Theorem (3.2) in Part I [7] ) we need

the minimal transitive permutation groups of degree 9. These are conjugate in Sg to

P1 ■= ((123456789)) a C9 and

P2 ■■= <(123)(456)(789), (147)(258X369)> *< C3 x C3

as one easily sees from the following lemma.

(2.1) Lemma. A minimal transitive permutation group P of prime power de-

gree pk is of prime power order.

Received November 27, 1978.

AMS (MOS) subject classifications (1970). Primary 20C10, 20H15.

Key words and phrases. Integral matrix groups.

*For n = 9 Q- and C-irreducibility of finite subgroups of GL(n, Z) coincide, as can be seen

by standard arguments using the results in ( 1 ].

© 1980 American Mathematical Society

0025-571 8/80/0000-001 6/$04.50

245

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

246 WILHELM PLESKEN AND MICHAEL POHST

Proof. Let S be a p-Sylow-subgroup of P and P0 the stabilizer of an element of

the set £2 on which P acts. We must only show that S also acts transitively on Í2

which is tantamount to SP0 = P. By counting cosets one obtains

\S\\P0\\SP0\ =

\snp0\

Since |S n P0\ divides \S\ as well as |P0|, we get |SP0| = |P|; hence SP0 = P. Q.E.D.

The methods employed in Parts I and II [7], [8] now yield:

(2.2) Lemma. 77ze solvable minimal irreducible finite subgroups of G¿(9, Z)

are rationally equivalent to

Gx =<diag(-1,1, 1,-1, 1,1, 1,1, \),D(PJ>, |G,| = 269 orto

G2 = <diag(-l, 1, 1,-1,-1, 1, 1,-1, \),D{P2)), \G2\ = 249.

Here and in the following D denotes the natural permutation representation of

S9, D: S9 —*■ GZ(9, Z) with Din)et = ejr(i) for rr E S9, i = 1, ..., 9, ev ..., e9 the

standard basis of Z9 * l.

Proof. Let G be a minimal irreducible subgroup of G£(9, Z), A the natural

representation of G, and N a maximal abelian normal subgroup of G. As in Part II [8]

we apply Theorem (3.1) of Part I [7]. Thus, we may assume that the restriction A\N

is equal to Tj + ■ • • + rr, where Tj, ..., Tr are integral representations of N satisfying

r\ ~Q ¿fcA,. (z = 1, ...,r;ifeeN)andr1(zV)= ■■■ = Tr(N). The A,, (z = 1, ...,r)are

inequivalent Q-irreducible integral representations of N all of the same degree m.

Since the Q-enveloping algebra of A^N) is a cyclotomic field, m has to be even

or equal to 1, hence m = 1 because of m\9. There remain three possible solutions of

the degree equation 9 = krm, namely (i) k = m = 1, r = 9, (ii) k - r = 3, m = 1,

(iii) m = r= l,k = 9.

Case (i). According to Theorem (3.2) in Part I [7] and Lemma (2.1) G is a sub-

group of the group generated by all diagonal matrices and DiP{) or by all diagonal

matrices and DiP2). Moreover, by the Schur-Zassenhaus Theorem G must split over

N. Again by Theorem (3.2) in Part I, N is minimal with the properties (a) N Ç

{diag(ap .... a9)\a¡ = ± 1, (i = 1, ..., 9)} and N is invariant under conjugation by

D(Pt) or DiP2), respectively; iß) the projections N —► {± 1} : diag(aj, ..., a9) —*-a{

(z = 1, ..., 9) are pairwise unequal. Then G can be chosen as (TV, Z)(P,)> or C/V, 73(y?2)>,

respectively. For £)(i'1 ) Ç G one easily sees that there is only one possibility for N,

whereas for DiP2) Q G there are six possible candidates for N. These, however, are all

conjugate under the normalizer of D(P2) in G£(9, Z). Hence, we end up with two

groups in Case (i).

Case (ii). Here Af must be a subgroup of index 1 or 2 of {diag(±/3, ±/3, ±/3)}.

Therefore, the nonzero quadratic forms which are fixed by G have matrices

diag(/4j, A2, A3) with symmetric definite A¡ E Z3 x3 (z = 1, 2, 3). G can be chosen

in such a way that A¡ = A2 = A3 and that G is a subgroup of H ^ S3, where H is

the automorphism group of Al [2]. Since G is minimal irreducible of odd degree, it can-


MAXIMAL FINITE IRREDUCIBLE SUBGROUPS OF GL(n, Z). Ill 247

not have a normal subgroup of index 2, hence G < H "\> C3. But all finite irreducible sub-

groups of GL(3, Z) are isomorphic to subgroups of C2 x S4. Thus, N cannot be a

maximal abelian normal subgroup of G; and there is no group in Case (ii).

Case (iii). Either N must be trivial and G, therefore, nonsolvable or N = <-/9>;

and G has a normal subgroup G n SZ,(9, Z) of index 2 and, hence, has a proper irre-

ducible subgroup. Q.E.D.

For the determination of the nonsolvable groups G we use the classification of

the primitive finite subgroups of 51(9, C) by Feit [3] and Huffman and Wales [4].

(2.3) Lemma. 77ze nonsolvable minimal irreducible finite subgroups of

GZ,(9, Z) are rationally equivalent to

G3 = <(e4, e8, e6, es, ev e9, e0, e3, e2), (e0, e9, ev c8, ev e2, e6, e3, es))

~A6=PSLi2,9).

As in Lemma (2.2) ex, ..., e9 denote the standard basis ofZ9xl,e0 = -2?_ j e¡.

Proof. We use the terminology of the proof of Lemma (2.2). Clearly, only

Case (iii) can occur. Since G must be contained in SZ,(9, Z), we have N = il).

We assume that G has a minimal normal subgroup M # < 1>. Then M is charac-

teristically simple, and A\N becomes (C-)reducible with 3-dimensional (C-)constituents.

A comparison with Blichfeldt's list of primitive subgroups of SL(3, C) [1] shows that

this is impossible because of rationality conditions for the characters. Therefore, G

must be simple and, hence, quasi-primitive. By [3] and [4] G has to be isomorphic

to A6 or A10. But the 9-dimensional irreducible representations of A6 and Al0 are

both obtained by reducing their permutation representations of degree 10 (note that

A6 = PSL(2, 9)). So G = Al0 is not minimal irreducible. Since PSL{2, 9) does not

have any doubly transitive subgroups acting on 10 points, A6 yields a minimal irreduc-

ible subgroup G3 of GZ,(9, Z). Q.E.D.

3. Computation of the Z-Classes in the 9-Dimensional Case. The centerings of

the minimal irreducible subgroups Gv G2, G3 of GL(9, Z) yield 20 quadratic forms

F, whose automorphism groups are the maximal finite irreducible subgroups of GL(9, Z).

Let Jn denote the n x n matrix with all entries equal to 1. Then the forms are given

by the following matrices F¡, their determinants by d{ Q — 1, ..., 20):

2 1 0 0 0 0 0 0 0\

121000000

fj ¿i = i; ^2 =

0 12 1

0 0 12

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

10 0 0

2 10 0

12 11

0 12 0

d7 = 4;

000000102/


WILHELM PLESKEN AND MICHAEL POHST

/

00000002

40000002

04000002

00400002

00040002

00004002

00000402

00000042

22222229

\

^3=4»

/3®(/3+/3), d4 = 43;

/3®(4/w3), d5=4';

dn = 47;

¡\


MAXIMAL FINITE IRREDUCIBLE SUBGROUPS OF GL(n, Z). Ill

c?9=4s;

Fl0 = (J3 + *3) ® V3 + J3)> d10=4416;

^i i = (4/3 -'3) ® (4/3 -'3). ¿1, = 44164;

12

/ 6 -2 -2 3-1 -1

/-2 6-2-1 3 -1-2 -2 6-1-1 3

3-1-1 6 -2 -2-1 3-1-2 6 -2

-1 -1 3-2-2 6

3-1-1 1 1 1

1-13-1111

\-l -1 3 1 1 1

'12

/

*13«

84400000

48400000

44800000

00084400

00048400

00044800

00000084

\00000048

\ 4 4444444

J13 = 46162

^14 = (/3 + ^3) ® (4'3 - '3), ¿14 = 4516¿ ;

Fls = 10/9-/9, d,5 = 108;

^16=/9+'/9. rfl6 = l°;



17

l-3 -3

/

^18 =

l 2 2 2

\ 0 0 0

/l2 2

/ 2 12

3

3

3

8

3

3

3

3

2

*"l9 =

2-3 3-3 2 3

2 -3 12 2 -2-3 2 2 12 3

3 3-2 3 12

3

3

3

3

3

8

3

3

2

:\

/

3 -3

3 -3 \

!

d18 = 2710;

3-3-2 3

3-3 3-2

3 2 3-2-2233

2 3 2 2

\

3 3 3-2 2 12 3-3-3-3-3 2 2 3 3 12 -2 3

-2 3 3 3 2-3-2

-2-2 3 2-3 312 -3 /

-3 12/

'17 5710;

'19 5310 • 204;

^20 ~

-2

1-1

0-1

1

1 -1 0-1 1

4 2

2 4

2 0

0 -1

0 0

1 0

2 0

0 -1

4 -1-1 4

0 1

2 0

\ 2-1 1-1-1 1

\-l 2 1 1-1-1

d = 2 • 4J • 20.



Now we describe the ■<-maximal centerings of the minimal irreducible groups Gv G2,

G3 which were computed by machine. If L is an irreducible ZG-representation module

and H a -<-maximal centering of L, then M* denotes the unique -<-maximal centering

of L which belongs to the inverse transposed representation coming from M [7].

Lattice of centerings for G1:

ic^k^ri

r# = j j# = j r# = iL'l ^l> ^2 L,V ^3 L,f,'

I* = I T# — J

(Read©=¿,(z= 1.9).)

Bases of the centering (numbers in brackets refer to the corresponding quadratic form

Ft)'(1) 5(1.) = /8,

(2) BiL2) = (x,, hxxv ..., h\xv v,) with x\ = (1, 1, 0, ..., 0), y\ =

, 0, 1, - 1) and Ä! = L\i 123456789)),(0,

(6) BiL3)

'o 1 1 0 0 0 0 0 l\

0 0 0 0 0 0 0 1 1

0 0 0 0 1 10 0 1

10 10 0 0 0 0 0

000000100

0 0 0 10 10 0 0

1 10 0 0 0 0 0 0

0 0 0 0 0 0 1 10

0 0 0 1 10 0 0 0,

(5) B(LA) = £>((24)(37)(68)) (/3 ® (/3 - 2/3)),

(4) BiL5) = Z)((24)(37)(68)) (73 9 (/3 - /3)),

(7) BiL6) = (2e,, 2e4, 2e7, 2e3, 2e6, 2e2, 2e5,x2,72) withx[ =

(0, 1, 1, 0, 1, 1, 0, 1, l),y¡ = (1, 1, 0, 1, 1, 0, 1,-1, 0) (the e, are defined in

(2-3)),



(3) fi(I7) = (2e1,...,2e8,-c0),

(12) BiU) =

0 0 0 1

0

0 0 0-1

0

0 0 0-1

0

0 0

0 0

\

:

(13) BiL9) = (2(e, + e4), 2(ex + e7), 2(e4 + e7), 2(e2 + es), 2(e2 + e8),

2(^5 + es)> 2(e3 + e6), 2(e3 + e9), ~e0).

r# =

We proceed to the -<-maximal centerings of G2 : Mx = Mx, M2 = M32, M3 = M31'

M26, M# = M25, M* = M24, Mf0 =M23,Ml = M30, M* = M29, Ml = M28, M.

M*x = M22, Mf2 = A/21, Mf3 = M20, Mf4 = M19, Mf5 = Mls, M#6 = M17, M2#7

= Af39, M33 = M3S, M34 = M37, M3S = M36. (See figure on next page.)

Because of the large number of centerings we do not give a list of all bases but

describe how to obtain them.

(i) Wi) = i9,

(2) BiM2) = BiL2),

(6) BiM3) = £((2734) (5896))fi(Z,3), BiM¡) (i = 4, 5, 6) are given by £>(7r)5(M3),

where ir is a suitable element of the normalizer of P2 in 59,

(3) BiM7) = I3® iJ3 -13), BiM¡) (i = 8, 9, 10) are obtained as in (6),

(8) BiMxl) i~ex + e2 + e3, ex - e2 + e3, ex+ e2 e4 + es + e6-

e5 + e6, ( 4 + e5 - e6, ex + e5 + e9, e2 + e6 + e7, e3 + e4 + e8), for 7?(A/()

(z = 12, ..., 16) compare (6),

/l 0 1 1

1110 1

WBiMxl)

!

0 110

0 0 0 1

0 0 0 0 1 10

0 0 0 10 11

0 1 10 0 0 0

0 10 0 0 1

10 0 0 0 1

0

1

1

1

1

0

1

0

1

0

1

1

0 0

0 0

0 0

1 1

:\

1

0

1

for BiM¡) (i = 18,

compare (6),

.,22)

:;/

(10) BiM21) = 0((123)(798))((/3 - 73) 9 (/3 - 73)),

(5) BiM26) = I39 iJ3 - 2/3), for 5(M,.) (z = 23, 24, 25) compare (6),



■<j\\

/ // x/ i \

w\

7 yy wX Vr<

(Read© = M,.(z = l,...,39).)

(7) 5(M31) = (2ex, 2e2, 2e3, 2e4, 2e5, 2e7, 2e8,x3,j'3) with x3 =

(0, 0, 0, 1, 1, 1, 1, 1, \),yT3 = (1, 1, 1, 0, 0, 0, 1, 1,-1), for B(M(.)(z = 28,29, 30)

compare (6),

(3) BiM32) = BiLn),

(12) 5(M33) = BiL8), BiM34) = TxBiM33) for a suitable monomial matrix Tx,



(14)

(13)

0 -1 -l\

1 0 1 1 ® (73 - 2/3),

1-1 0/

BiM3S) = T2 • 5(M36) for a suitable monomial matrix T2,

BiM3S) = BiL9),

BiM31) = T3 ■ BiM3S) for a suitable monomial matrix T3.

At last we describe the -<-maximal centerings of G3 : Nf = N6, N2 = N3, N* = Ns,

N# = NT

\

(Read© = ^,.(z= 1,...,8).)

(15) BiNx) = I9,

(17) 5(A/2) = (ej + e2, ex + e3, ..., ex + e9, -e2 - e3),

(18) BiN3) = i~ex + e2, -ex + e3, ..., -ex + e9, e5 + e6 + e7 + e8 + e9),

(19) 5(iV4)

/l 1 1 1 1 1 0 1 o\

/l 10 0 1 1 0 0 1 \

0 0 10 11110

0 0 0 11110 1

0 0 110 0 0 11

10 0 0 10 0 1 1

0 10 0 0 10 10

\ 1 0 1 0 0 1 1 0 1 /

\ 0 1 0 1 10 1 1 1 /

BiNs) = £>((1475)(2698))5(W4),

(16) B(N6) = I9+J9,



/

(20) BiN,) =

0 0 0 0 0

10 0 0 1

110 0 0

0-1 0 0-1

10 0 0-1

11110

0 -1 -1 -1 1

0 1 1

0 1 0

0 0-1

0 0 0

0 0 1

1 0 0

\

1-10 0

\ 0 0-1 1 0-1 0-1 1 /

\o 0 1-1 0-1-1 0 l/

BiN8) = £>((1475)(2698))£(/V7).

4. The Irreducible Maximal Finite Subgroups of GL(9, Z). There are—up to Z-

equivalence—20 such groups. They fall into eight Q-classes. Fourteen of these groups

(belonging to six Q-classes) are rationally equivalent to a monomial group and nine (be-

longing to three Q-classes) are nonsolvable.

(4.1) Theorem. 77ze irreducible maximal finite subgroups o/G¿(9, Z) are Z-

equivalent to the Z-automorphism groups of the quadratic forms Fx, ..., F20.

(i) Aut(Fj), Aut(F2), and Aut(F3) are ^-equivalent. They are isomorphic to

the wreath product C2 "V S9 of order 299!. Aut(Fj) is the full monomial group of

degree 9.

(ii) Aut(F4), Aut(F5), Aut(F6), and Aut(F7) are ^equivalent and isomorphic

to the wreath product (C2 ~ S3) -v S3 i=iC2 x S4) 'v S3) of order (233!)33!.

(iii) Aut(F8) ~q Aut(F9). Both are isomorphic to a split extension of an ele-

mentary abelian group of order 29 by a wreath product (53 'v C2). The order is

29(3!)22.

(iv) Aut(F10) ~q Aut(Flt). They are isomorphic to C2 x (54 'v C2) of order

2(4!)22. (77ze groups can be considered as crown products of C2 x S4 by C2.)

(v) Aut(F12) ~q Aut(F13). 77ze groups are isomorphic to C2 x (54 'v- S3) of

order 2(4!)33!.

(vi) Aut(F14) is isomorphic to C2 x 54 x S4 of order 2(4!)2.

(vii) Aut(Fj s), Aut(Fj 6), Aut(Fj 7), and Aut(Fj 8) are ^-equivalent and iso-

morphic to C2 x S10 of order 2(10!).

(viii) Aut(F19) ~q Aut(F20). Both are isomorphic to C2 x S6 of order 2(6!).

For a better understanding of the cases (ii), (iv), and (v) we note that the irre-

ducible maximal finite subgroups of G¿(3, Z) are all rationally equivalent and iso-

morphic to C2 'v S3 i=C2 x S4).

Proof. Ad(i). Compare Theorems (6.1), (6.2) in Part I [7].

Ad(ii). We have 4F4i - F5 and 4F¿"X ~z F?. Hence, by transposing the

matrices of Autz(F4) (Autz(F6)) one obtains a group which is Z-equivalent to Autz(F5)

(Autz(F7)). Since F4 = (/3 + 73) © (/3 + J3) © Q3 + J3), Autz(F4) is the wreath



product of Autz(73 + /3) with 53 [2], Autz(73 + J3) being isomorphic to C2 ^ S3.

It remains to show that Autz(F4) and Autz(F6) are rationally equivalent. In the lat-

tice of centerings for G2 the centerings Af7, M3 belong to the forms F4, F6, respectively.

Let G be the biggest subgroup of Aut(Af7) which fixes the quadratic form belonging to

Mn (i.e. G = Autz(F4)). We prove that G is rationally equivalent to a subgroup of

Autz(F6) by showing that G leaves M3 invariant. Clearly, G leaves M26 = M* invari-

ant and, hence, also Mx = Mn + M*. But even the maximal finite subgroup of

G¿(9, Z) belonging to Mx and Fx (i.e. the full monomial group of degree 9) leaves

M32 invariant. Hence, G leaves M3 = M7 + M32 invariant. Similarly, one sees that

Autz(F6) is rationally equivalent to a subgroup of Autz(F4) (note Mn = M2 n M3).

Ad(iii). The forms ^F^1 and F9 are integrally equivalent. Therefore, the corre-

sponding automorphism groups are Q-equivalent. We determine Autz(F8). In the

lattice of centerings for G2 for instance Mxx belongs to F8; hence, the biggest sub-

group G of Aut(Mn) fixing the corresponding form is isomorphic to Autz(F8). G

also leaves Mxx = M22, hence, Mx = Mxx + M22 invariant. Thus, G is Q-equivalent

to a subgroup G of the full monomial group, more precisely to the biggest subgroup

of Aut(Mj) which leaves Mx x and the form belonging to Mx invariant. Clearly, G

contains the minimal irreducible group G2, if we identify Aut(Mx) with GZ,(9, Z) via

the canonical basis. Furthermore, G contains all diagonal matrices of GZ,(9, Z), since

they fix MX/2MX pointwise. It follows that the diagonal matrices form a normal sub-

group of G of order 29, having a transitive permutation group P of degree 9 as com-

plement. Considering the vectors of shortest length in Mx x we find that P has to be

the maximal subgroup of S9 permuting the sets {1, 2, 3}, {4, 5, 6}, {7, 8, 9},

{1, 5, 9}, {2, 6, 7}, {3, 4, 8}. By a short computation P acts faithful and imprimi-

tive on these six sets and is isomorphic to S3 ^ C2.

Ad(iv). The forms 16FXq and Fxx are integrally equivalent, hence, Autz(F10)

~q Autz(F11). We determine the automorphism group of F10 = (/3 + J3) ®

(73 + J3). Clearly, Autz(Fj 0) contains all g ® h with g, h E Autz(/3 + /3) = C2 x S4

as well as an involution interchanging the components of the tensor product. But

these elements already generate Autz(F10). This is shown by similar arguments as in

(ii) or (iii); in particular, it follows easily that Autz(F10) is rationally equivalent to a

subgroup of Autz(F8). The index can be computed as in case (v) below.

Ad(v). Again the forms 16Fj"2 and F13 are integrally equivalent; hence,

Autz(F12)~Q Autz(F13). We determine Autz(F12). In the lattice of centerings for

G2 the centering M3 3 belongs to Fx 2. Let G be the biggest subgroup of Aut(M3 3)

fixing the corresponding form, G = Autz(F12). Then G leaves M23 = M33 + M33

invariant. Hence, G is isomorphic to a subgroup of Autz(F5) = (C2 x S4) 'v S3. Let

G be the maximal subgroup of Aut(M33) fixing F5. The group G acts on {gM33\gEG}

=■ S, transitively. The set S consists of four elements. Then G is isomorphic to the

stabilizer of M33 in G. The latter turns out to be isomorphic to C2 x (iS"4 "* S3).

Ad(vi). The automorphism group of the form F14 = (73 + J3) ® (4/3 - J3)

certainly contains the elements g ® h with g E Autz(/3 + J3) — C2 x S4> h E



Autz(4/3 ~/3) = C2 x S4. These form already the full automorphism group as can

be seen similar to the previous cases.

Ad(vii). It is lOFj-1 = F16 and lOFj",1 ~z F18. Therefore, Autz(Fls) ~Q

Autz(F16) and Autz(F17) ~Q Autz(F18). Since the representation module Nx of

G3 has—up to sign—only ten vectors of minimal length with not any two of them being

orthogonal, the group Autz(Fls) is easily seen to be isomorphic to C2 x 510 (com-

pare also [6]). To prove that Autz(F15) is rationally equivalent to a subgroup of

Autz(F17) we consider the lattice of centerings of Autz(Fls) viewed as a subgroup

of Aut^Vj). It consists of the multiples of Nx, Ns = Nx#, N2 = N* + 2NX,N3 =N#

([6], [7]). Therefore, N2 is invariant under Autz(Fls). On the other hand Nx =

N7 + N2 holds and Autz(F17) is rationally equivalent to a subgroup of Autz(F15),

hence to Autz(Fls) itself.

Ad(viii). The forms 20Fj-9 and F20 are integrally equivalent; and hence,

Autz(F19) ~q Autz(F20). In the lattice of centerings for G3 the centering N5 be-

longs to the form Fx 9. Because of N5 + N* = N2 and N2 + N* = Nx the group

Autz(F19) is rationally equivalent to a subgroup of Autz(F15) = C2 x Sx0. There-

fore, Autz(F19) is isomorphic to C2 x P, where F is a subgroup of Sxo maximal with

the properties:

(a) P contains FSZ(2, 9) acting on the ten points of the projective line over F9 ;

compare Lemma (2.3).

iß) The natural permutation representation of P has two constituents unequal 1

modulo 2, each of degree 4.

From the list of primitive permutation groups up to degree 20 [9] F has to be a

subgroup of Aut(FSZ(2, 9)) s FrX(2, 9), since Axo and Sxo violate property iß). The

decomposition numbers modulo 2 of S6 show that the subgroup of Fr¿(2, 9) which

is isomorphic to S6 still fulfills (a) and iß) [5]. To prove S6 = F it remains to show

that iß) does not hold for FrX(2, 9). But iß) implies that F is isomorphic to a sub-

group of GZ,(4, 2), and the assumption F = PTL{2, 9) would lead to a primitive per-

mutation representation of GL(4, 2) of degree |GZ,(4, 2)|/|FrX(2, 9)| = 14 which does

not exist by inspection of Table I in [9]. Q.E.D.

RWTH Aachen

Lehrstuhl D für Mathematik

Templergraben 64

5100 Aachen, West Germany

Mathematisches Institut der Universität zu Köln

Weyertal 86-90

5000 Köln 41, West Germany

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iii. the nine dimensional case

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