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