12

MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN

Ei

=

t

for odd i, and choose

t2,

t4, ... , te

so that each element of

I"- {

t}

occurs

in this sequence exactly

n'

times.

Let g

E

G, and let

n

9

(u)

be the number of indices 1 :::; i :::;

e

for which

ai

=g.

From the above,

4ln9

(u). Moreover, a2i-l

=

t(tt2i)t

=

a;i~

1

=

a2i, for each

1:::;

i:::;

e/2. Hence,

n

9

(u) = n

9

-1(u) = n

9

.(u) = n

9

-,(u).

This yields (d1).

C. Construction

ofT.

For each [C]let

V[CJ

be a positive integer. Choose

k and

g =

(g1,

... ,gk) E (;k

such that

nc(g) =

V[CJ•

for each C. In particu-

lar, g contains an entry from each C

E

£(G). A proper subgroup of G misses

some conjugacy classes of

G

[FJ, Lemma 12.4]. Therefore,

G

= (g1

, ...

,gk)·

Furthermore, k

=

l::cE.C(G) V[c]

=

!!j.

Put

(

-1 -1

E -E E -E)

T

=

91,91 '· · ·'

9k.9k ,gk,gk '· · ·

,g1,gl ·

This choice satisfies (a2) and (c2), and, for each C,

nc(-r) = nc(g) + nc-1 (g)+ nc• (g)+ nc-· (g)= 4v[c]· o

LEMMA

3. 2.

Let

1 ___,

G ___, H

~C

___,

1

be

an exact

sequence

of

finite groups,

and

let I be

a

set

of

involutions in H"-G. There exists

a

commutative diagram

1-G-ii----=--C-1

7r

(1)

II

1 - G - H

----,;:---+-

C - 1

with

exact

rows

and

surjective vertical maps such that the Schur multiplier

of

G

is generated by commutators

and

C

if

(G)

=

1.

Finally, every involution in I

lifts to

at

least two involutions in

fi.

PROOF. Choose a presentation 1

---f

n

---f

:F

---f

H

---f

1, where

:F

is the free

product of a free group of finite rank with finitely many groups of order 2, say

(81), ... , (8e),

such that

{81, ... ,8e}

maps onto

I.

The inverse image

F1

of

G

in

:F

contains no conjugates of 81,

... ,

8e.

By the Kurosh Subgroup Theorem

[M, Theorem VII.5.1 and Proposition VII.5.3] it is a free of finite rank. Let

N

=

[:F1,

n]

be the group generated by commutators

[f,

r] with

f

E

:Fl, r

En.

Set

F = :F/N,

F

1

= :FI/N,

and R

= RjN.

Then 1 ___, R ___, F

1

___,

G ___,

1 is a

central extension.

Schur multiplier theory [Hu, Kap.5, §23] shows R is the direct product of

the Schur multiplier

M(G)

=

R

n

(FI)'

and a free abelian group

A.

Let

Ao

be

the intersection of all the F-conjugates of

A.

Then

Ao

1

F.

Since

(R : A)

=

IM(G)I

oo, also

(F: A

0 )

oo. Set

fi

= F/Ao,

G

= FI/Ao,

and

S = R/Ao,

to get diagram (1). The image

l

of {

81

, ... ,

8e}

in

fi

maps onto

I.

Notice that

Sis the direct product of

S

n

(G)'

~

M(

G) and

A/ Ao.

As in the proof of

[FV3,

Lemma 2] the Schur multiplier of

G

is generated by commutators.

Replace

G, H,

and

I

by

G,

fi,

and

Y,

to assume that the Schur multiplier

of

G

is generated by commutators. Let

T

be a non-abelian finite simple group

with trivial Schur multiplier. For example, take

T

=

SL2(8)

[Hu,

Satz 25.7].