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].
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