REAL HILBERTIANITY-TOTALLY REAL NUMBERS
25
set of involutions in
G.
A morphism of involutory structures ¢: G ---+ H is a
continuous homomorphism of groups¢:
G---+
H such that
¢(Ic)
~
IH.
We say
that¢: G---+ His an epimorphism if ¢(G)= Hand
¢(Ic)
=
IH.
EXAMPLE 7.1 (A). Let L
I
K be a Galois extension with L not formally real.
Then
G(LIK)
=
(G(LIK),I(LIK))
is an involutory structure. Let
E
be an
extension of
K,
and let
FIE
be a Galois extension such that
L
~
F.
Then the
restriction resL:
G(FIE)
---+
G(LIK)
is a morphism. Moreover, suppose that
E
I
K
is regular and totally real: every ordering on
K
extends to
E.
Then resL
is an epimorphism
(cf.
[HJl, Lemma 3.5]).
EXAMPLE 7.1
(B).
Let
SIR
be a real Galois cover with
FIE
the correspond-
ing Galois extension of fields. Let
M
be a field and let}!_:
R
---+
M
be a ho-
momorphism. Extend ¢ to a homomorphism 'lj;:
S
---+
M.
Then the group
homomorphism 'lj;*:
G(M)
---+
G(F I E)
is a morphism of involutory structures
'lj;*: G(M) ---+
G(FI E)
(Remark 6.2(b)).
A finite image of G is a finite involutory structure H for which there exists
an epimorphism ¢: G ---+ H. Clearly, up to an isomorphism, it is of the form
(GIN, {ENINI
E E
Ic}),
where
N
is an open normal subgroup of
G
not meeting
Ic.
Let ImG be the class of all finite images of G.
A finite embedding problem for G consists of an epimorphism
Jr:
H---+
A
of finite involutory structures, together with an epimorphism ¢: G ---+ A. A
solution is an epimorphism 'lj;: G ---+ H such that
1l'
o 'lj; = ¢. We say that
G has the embedding property if every finite embedding problem (
Jr:
H ---+
A,
¢: G ---+
A)
for G, in which H is a finite image of G, has a solution.
EXAMPLE 7.2. Let
D
be the free profinite product
fixEXw
(Ex) of groups of
order 2 over
Xw
(Remark 1.5), and let
Ifjy
= {Exl x E
Xw}·
A finite involutory
structure (A,
IA)
is a finite image of (D,
Ifjy)
if and only if A is generated by
IA.
Furthermore,
(D, Ifjy)
has the embedding property.
With
D
and
Ifjy
as above, put D =
(D, ID),
where
ID
is the conjugacy domain
ConD(Ifjy) of D generated by
Ifjy.
LEMMA 7.3. (a)
ID is all involutions in D, and Dis of rank
~ 0 .
(b)
A finite involutory structure
A
is in
ImD
if and only
if
IA
=1-
0
is a
conjugacy domain in A and A= (IA)·
(c)
D
has the embedding property.
PROOF OF (a). See [HJ2, Corollary 3.2 and Lemma 2.2]
PROOF OF (b). Immediate from Example 7.2.
PROOF OF (c). Let
Jr:
H---+
A,
¢: D---+
A
be a finite embedding problem for
D. Then
IA
~
A and
IH
~
Hare conjugacy domains. Let
I~
=
¢(Ifjy)
and let
I~
= {E E IHJK(E) E
I~}.
As ConD(Ifjy) =
ID,
we have
ConA(J~)
=
IA;
it
follows that Con
H
(I~)
=
I
H.
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