REAL HILBERTIANITY-TOTALLY REAL NUMBERS 13
Form the regular wreath product
ii
of H with T (e.g.,
[Hu,
Def.
15.6]).
Thus
ii
=
TJ JH,
with j
=
I HI,
and
H
acts on
TJ
by permuting the factors in its
regular representation. Let
G
be TJ
JG:::;
H.
Clearly,
C
fi(TJ)
=
1, and hence
Cfi(G) = 1. If
E
E
I and
T
E
TJ,
then
T-
1
ET
is an involution in
ii
that maps to
E.
This proves the last assertion
of
the lemma.
Since
M(T)
=
1, every central extension
ofT
splits. This implies that every
representation group of
G
has a normal subgroup isomorphic to TJ such that the
quotient by this subgroup is a representation group of
G.
Therefore,
M(G)
~
M( G) is generated by commutators. o
LEMMA
3.3.
Let 1r: H
--+
H be an epimorphism of finite groups, and let
h, ... ,
Im
~Hand
/1, ... ,
Im
~
H be
sets
of
in~lutions
such that 1r(Ij)
=
Ij.
Then there exists a finite
group
H, a surjection
p:
H
--+
H, and sets of involutions
h_, ... ,
Im
~
H
such that
p(~)
=
Ij for every
j,
and every automorphism
Ci
of
H that satisfies a(lj)
=
Ij for all
j,
lifts to an automorphism
a
of
ii
(that is,
(1r
o
p)
o
a= a
o
(1r
o
p)) that satisfies_ a(~)=~ for all
j.
Moreover, if the
!j
are conjugacy domains in H, then the Ij can be taken conjugacy domains in H.
PROOF.
Let
K
be a set of cardinality Ker(
1r).
Consider the free product
H
= (
H
(xx,k))
* (
H
(E,k,l))
* (
H
(E£,k,2))
* · · · * (
H
(E,k,m) ),
hEll
kEK
of cyclic groups. Here
(xx,k) ~
Z and
(E,k,j) ~
Z/2Z, and let
~
=
{f,,k,j
I
t
E
Ij,
k
E
K}.
(Of course,
H
is not yet finite.) Define a surjection p:
ii
--+
H
by mapping
{xx,kl
k
E
K}
onto
{h
E
HI
1r(h)
= h}
and
{E,k,jl t
E
Ij}
onto
{~:
E
Ijl7r(E)
=
E}.
Then
p(~)
= Ij. Every automorphism
a
of H that satisfies
a(lj)
=
Ij
for all j, lifts to an automorphism
a
of
ii
defined by
xh,k
f-+
xa(h),k
and E,k,j
f-+
Ea(!_),k,j.
Clearly
a(~)
=
~.
If Ij and
Ij
are conj':gacy domains,
we can replace Ij by the conjugacy domain that it generates in H.
Thus
ii
satisfies the requirements of the lemma, except that it is not finite.
To make
ii
finite, replace it by its quotient
H /
N,
and p by the ir:duced quotient
map, where N is a characteristic subgroup of finite index in H, contained in
Ker(p). For example, takeN to be the intersection of all normal subgroups M
of
ii
with
H/M
~H.
o
4. Points over ordered fields
Let
G
be a finite group with a trivial center such that the Schur multiplier of
G
is generated by commutators. Identify
G
with the subgroup Inn(
G)
of Aut(
G).
Fix a sufficiently large integer
r
that satisfies (2.13) and the assertions of Lemma
3.1. Associate with
G
and
r
the moduli spaces
11in
and
11ab
from (2.3).
Our aim is to choose Hurwitz spaces
1i
and if and some points q = (b, 0,
fo]
on
1i
as in (2.4). First, let
e
=
8 ·
IGI!
and
m
=
r;e,
so
r
=
e
+2m. Define the
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