Large localizations of finite simple groups
R"udiger G"obel, Jose L. Rodriguez and Saharon Shelah *
Abstract
A group homomorphism j : H ! G is called a localization of H if every
homomorphism ' : H ! G can be `extended uniquely' to a homomorphism
: G ! G in the sense that j = '.
Libman showed that a localization of a finite group need not be finite*
*. This
is exemplified by a well-known representation An ! SOn-1(R) of the altern*
*ating
group An, which turns out to be a localization for n even and n 10. Emma*
*nuel
Farjoun asked if there is any upper bound in cardinality for localization*
*s of An.
In this paper we answer this question and prove, under the generalized co*
*ntinuum
hypothesis, that every non abelian finite simple group H, has arbitrarily*
* large
localizations. This shows that there is a proper class of distinct homoto*
*py types
which are localizations of a given Eilenberg-Mac Lane space K(H; 1) for a*
*ny non
abelian finite simple group H.
0 Introduction
One of the current problems in localization of groups is to decide what algebra*
*ic prop-
erties of H can be transferred to G by a localization j : H ! G. Recall that j *
*: H ! G
is a localization if every homomorphism ' : H ! G in the diagram
j
H -! G
' # . (*
*0.1)
G
___________________________________
*The authors are supported by the project No. G 0545-173, 06/97 of the Germa*
*n-Israeli Foundation
for Scientific Research & Development.
The second author is also supported by DGES grant PB97-0202 and DAAD.
GbSh 701 in Shelah's list of publications.
1
can be extended to a unique homomorphism : G ! G such that j = '. This in
other words says that G ~= LjH where Lj is the localization functor with respec*
*t to
j; see e.g. [2, 13, 4, 3]. For example, the properties of being an abelian gr*
*oup or a
commutative ring with 1 are preserved. Casacuberta, Rodriguez and Tai [4] have *
*found
consequences of these facts for homotopical localizations of abelian Eilenberg-*
*Mac Lane
spaces, see also Casacuberta [3].
A further step leads to nilpotent groups. Dwyer and Farjoun showed that ev*
*ery
localization of a nilpotent group of class 2 is nilpotent of class 2 or less. *
*A proof is
given by Libman in [13] (see also [3]). However, it is unknown if nilpotent gro*
*ups are
preserved under localizations in general.
Another interesting problem is to find an upper bound for the cardinalities*
* of the
localizations of a fixed group H. It is easy to see that, if H is finite abelia*
*n, then every
localization j : H ! G is an epimorphism, hence |G| |H|. More generally, Libm*
*an
has shown in [13] that if H is torsion abelian then |G| |H|@0. However, if H *
*is not
torsion, this can fail. Indeed, the localizations of Z are precisely the E-rin*
*gs, and we
know by Dugas, Mader and Vinsonhaler [9] that there exist E-rings of arbitraril*
*y large
cardinality. Str"ungmann sharpened this result in [22] for almost free E-rings.
The first example observing that a localization of a finite group need not *
*be finite is
due to Libman [14]. He showed that the alternating group An has a (n - 1)-dimen*
*sional
irreducible representation j : An ! SOn-1(R) which is a localization for any ev*
*en
natural number n 10. In the proof he uses that On-1(R) is complete, SOn-1(R)
is simple, and the fact that all automorphisms of SOn-1(R) are conjugation by s*
*ome
element in On-1(R). This also motivates our Definition 1. Thus, Emmanuel Farj*
*oun
asked about the existence of an upper bound for the cardinality of localization*
*s of An.
We give an answer to this question in Corollary 3, which is a direct consequenc*
*e of our
Main Theorem and Proposition 2.
In fact our result also holds for many other finite groups which we will ca*
*ll suitable
groups, see Definition 1. We shall assume the generalized continuum hypothesis.*
* GCH
will be needed to apply a new combinatorial principle which is similar to `Shel*
*ah's black
box' or the diamond principle }. But rather than applying some game with a winn*
*ing
strategy for some player we will apply the outcome directly as stated in Propos*
*ition 5.3.
The proof of this combinatorial result will appear in Chapter 8 of the book by *
*Shelah
[21]. The proof can also be recovered from [11], the result is stated for cardi*
*nality @1 in
2
[20] and applied to boolean algebras, moreover see [19]. Accordingly, the group*
* G being
constructed will have cardinality |G| = + , the successor of a regular cardinal*
* .
The group theoretical techniques derive from combinatorial group theory and*
* can
be found in the book by Lyndon and Schupp [15]. Some aspects are inspired from
the solution of Kurosh's problem about Jonsson groups [18]. But unlike there, i*
*n this
updated version we will not apply cancellation theory which simplifies proofs a*
*bout
certain centralizers of subgroups.
Main Theorem (ZFC + GCH) Let be the successor of an uncountable, regular
cardinal and let + be its successor cardinal. Then any suitable group H is a su*
*bgroup
of a group G of size + with the following properties:
(a) Any monomorphism OE : H ! G is the restriction of an inner automorphism o*
*f G.
(b) H has trivial centralizer in G: if [H; x] = 1 for some x 2 G then x = 1.
(c)Any monomorphism G ! G is an inner automorphism of G.
(d) G is simple.
It is interesting to note that we can also require that G = G[H], this is t*
*o say that
the following holds.
(e) The group G is generated by copies of H.
This strong demand can be established if we add G[H] = G to the Definition *
*3.2 of
K*. It is then easy to see from Lemma 3.11 that the new class K* is still large*
* enough
to provide G as in the Main Theorem. We see that the group G in the Main Theore*
*m is
complete, i.e. has trivial center Z(G) and Aut (G) = Inn(G), where Aut (G) deno*
*tes the
automorphism group of G and Inn (G) is the normal subgroup Inn (G) = {g* : g 2 *
*G}
which consists of all conjugations
g* : G -! G (x -! g-1xg):
It is also obvious that G from the Main Theorem is co-hopfian in the sense that*
* any
monomorphism is an automorphism of G.
Definition 1 Let H be any group with trivial center and view H Aut (H) as inner
automorphisms of H. Then H is called suitable if the following conditions hold:
3
1. H is finite and Aut (H) is complete.
2. If H1 Aut(H) and H1 ~=H then H1 = H.
Note that Aut (H) has trivial center because H has trivial center. Hence t*
*he first
condition only requires that Aut (H) = Inn(H), so H has no outer automorphisms.*
* We
want to show that all non abelian finite simple groups are suitable. As a conse*
*quence
of the classification of finite simple groups the Schreier conjecture holds for*
* all finite
groups H, hence the outer automorphism group Out (H) ~=Aut (H)=Inn (H) is always
soluble. As H is also simple and non abelian then H identified with Inn (H) is*
* a
characteristic subgroup of Aut (H). Hence, looking at the solvable group Aut (*
*H)=H
any copy of H in Aut (H) must be H. Moreover Aut (H) is complete by Burnside, s*
*ee
[16, p. 399]. This shows part (a) of the following
Proposition 2 (a) All non abelian finite simple groups are suitable.
(b) The non abelian, finite, simple and complete groups are precisely the follo*
*wing spo-
radic groups: M11; M23; M24; Co3; Co2; Co1; F i23; T h; B; M; J1; Ly; Ru; J4 a*
*nd the
following Chevalley groups for primes p and natural numbers n 3
S2n(2); G2(p) (p 6= 2; 3); F4(p) (p 6= 2); E7(2); E8(p):
Part (b) follows by inspection of the list of finite simple groups. It is int*
*eresting to
know when H = Aut(H) because in this case our proof becomes visibly simpler.
We finally obtain an answer to Emmanuel Farjoun's question concerning alter*
*nating
groups An for all finite simple non abelian groups.
Corollary 3 (Assume ZFC + GCH.) Any finite simple non abelian group has localiz*
*a-
tions of arbitrarily large cardinality.
The localization An ! SOn-1(R) for any even n 10 induces a map between
Eilenberg-Mac Lane spaces K(An; 1) ! K(SOn-1(R); 1) which turns out to be a lo-
calization in the homotopy category. This is the first example of a space with*
* a fi-
nite fundamental group which admits localizations with an infinite fundamental *
*group.
Corollary 3 yields then the following extension.
Corollary 4 (Assume ZFC + GCH.) Let H be a finite simple non abelian group. Then
K(H; 1) has localizations with arbitrarily large fundamental group.
Constructions in homotopy theory based on large-cardinal principles were us*
*ed in
Casacuberta, Scevenels and Smith [5].
4
Acknowledgements. We would like to thank Emmanuel Farjoun for explaining the
above problem to us and pointing out further interesting connections between gr*
*oup
theory and homotopy theory.
1 Proof of corollaries
Assuming GCH, Proposition 5.3 applies for any cardinal + with uncountable and
regular, hence G in the Main Theorem can have cardinality + . In order to prove
Corollary 3, we next show that the inclusion j : H ,! G in the Main Theorem is a
localization. Suppose that ' : H ! G is a homomorphism. We have to show that th*
*ere
is a unique homomorphism : G ! G such that j = '. If ' = 0 then = 0 makes
the diagram (0.1) commutative. To see that it is unique, we note that H is in t*
*he kernel
K of and K = G by simplicity, hence = 0. Now suppose that ' 6= 0. Since H is
simple we have that ' is a monomorphism thus by (a) of the Main Theorem there is
an element y 2 G such that ' = y* H where y* H denotes the restriction of the
map y* on H. Hence = y* satisfies j = '. Suppose that 0 : G ! G is another
homomorphism such that 0j = '. Then 06= 0 and since G is simple by (d), the map
0is a monomorphism of G, hence an inner automorphism by (c) . Now (g-1y)* : G !*
* G
fixes all elements of H. From (b), we obtain g-1y = 1 and thus = 0as desired.
Recall from [7] or [4] that a map f : X ! Y between two connected spaces i*
*s a
homotopical localization if Y is f-local, i.e. if the map of pointed function s*
*paces
map* (Y; Y ) ! map* (X; Y )
induced by composition by f is a weak homotopy equivalence. As in the case of g*
*roups
this says that Y ' LfX, where Lf is the localization functor with respect to f.
It turns out that the homotopical localizations of the circle S1 = K(Z; 1) *
*are pre-
cisely Eilenberg-Mac Lane spaces K(A; 1) where A ranges over the class of all E*
*-rings
[4, Theorem 5.11]. And therefore this is proper class (not a set) in view of th*
*e result in
[9]. Recall that an E-ring is a commutative ring A with identity which is cano*
*nically
isomorphic to its own ring of additive endomorphisms. Corollary 4 claims that a*
* similar
situation holds for K(H; 1) if H is a finite, simple non abelian group. However*
*, in this
case, other localizations of K(H; 1) which are not of the form K(G; 1) may exis*
*t, see [7,
Section 1.E.].
5
Corollary 4 follows from the fact that every localization of groups H ! G g*
*ives rise
to a localization of spaces K(H; 1) ! K(G; 1). This holds because, for arbitrar*
*y groups
A and B, the space map* (K(A; 1); K(B; 1)) is homotopically discrete and equiva*
*lent to
the set Hom (A; B).
Basic facts on homotopy theory can be seen in [23], the monograph by Aubry *
*[1]
(lectures of a DMV seminar by Baues, Halperin and Lemaire) as well as in Farjou*
*n's
exposition [7] for homotopical localizations.
2 On free products with amalgamation and HNN
extensions
If G is a group and a; b 2 G then [a; b] = a-1b-1ab denotes the commutator of a*
* and b,
and this naturally extends to subset [A; B]. Compare Theorem 2.7, p. 187, Theor*
*em 6.6,
p. 212 and Theorem 2.4, p. 185 in [15] for the notion of free products with ama*
*lgamation
and HNN extension. We will need a lemma describing finite subgroups of free pro*
*ducts
with amalgamations and HNN extensions.
Lemma 2.1 Let G* = G1 *G0 G2 be the free product of G1 and G2 amalgamating a
common subgroup G0 = G1 \ G2 and let H be a finite subgroup of G*. Then there e*
*xist
i 2 {1; 2} and y 2 G* such that (H0)y Gi.
Proof. Let
* x *
HG = {H : x 2 G }
be the conjugacy class of H in G*. If g 2 G* = G1 *G0 G2 then |g| denotes the l*
*ength of
*
g, which is an invariant of g, see [15]. We now choose H0 2 HG subject to the *
*following
two conditions.
|H0\ (G1 [ G2)| is maximal (see also (2.5)) (*
*2.1)
and among those let
min{|h| : h 2 H0\ G1 \ G2} be minimal, also saymin ; = 0: (*
*2.2)
So there is such an H0 which we rename H. If h 2 H, then h is torsion and b*
*y the
Torsion Theorem for free products with amalgamation there are g 2 G* and i 2 {0*
*; 1}
6
such that hg = y 2 Gi, see [15] or [16]. Hence any h 2 H has the form
h = gyg-1 = g1. .g.nyg-1n. .g.-11: (*
*2.3)
and we may assume that n minimal with (2.3). If y 2 G0, then we can replace y *
*by
gnyg-1n, which is in G1[ G2 and g1. .g.n-1has shorter length, contradicting min*
*imality
of n. Hence y 2 G1 [ G2 \ G0 and say y 2 G1 without any restriction. By the s*
*ame
argument we can not have that gn 2 G1, hence gn 2 G2 and h = g1. .g.nyg-1n. .g.*
*-11is
in normal form and its length is |h| = 2n + 1 is odd. We have shown that
|h| is odd for all elementsh 2 H: (*
*2.4)
If hi 2 H \ Gi\ G0 for i = 1; 2 then visibly h = h1h2 has length |h| = 2 wh*
*ich
contradicts (2.4). We conclude
H \ (G1 [ G2) = H \ Gi for somei 2 {1; 2}; (*
*2.5)
hence in (2.1) either |H \ G1| or |H \ G2| is the maximal integer. By symmetry *
*we may
assume that
H \ (G1 [ G2) = H \ G1 and hence |H \ G2| |H \ G1|: (*
*2.6)
In order to say more about H \ G1 we fix a left coset representation of G0 *
* Gi and
let 1 2 Zi Gi be a fixed left transversal of G0 in Gi, i.e. Gi is the disjoint*
* union of
{zG0 : z 2 Zi} and Z = Z1 [ Z2 is a transversal of G* over G0. Following stand*
*ard
notation if g 2 zG0 we also write g = z for the representative of the coset. We*
* will use
the following well-known fact about normal forms with respect to transversals; *
*see [16,
pp. 179-181] or [15, pp. 205-206]. Any
g 2 G* can uniquely be expressed as a reduced wordg = g1. .g.ng0 (*
*2.7)
with g0 2 G0 and gk 2 G1[ G2\ G0 alternating in G1 and G2 respectively, e.g. by*
* using
[16, p. 179]. If we apply this to (2.3) then (after renaming y as g0yg0-1 ) t*
*hen (2.3)
becomes an expression with unique gi2 Z
h = g1. .g.nyg-1n. .g.-11withgi2 Z: (*
*2.8)
Next we claim that
H \ G1 6 G0: (*
*2.9)
7
Assume H \ G1 G0 and also assume that the lemma does not hold. Hence there is
h 2 H \ G1 \ G2, which can be expressed as in (2.8). Now we claim that the subg*
*roup
*
H0 = Hg1 2 HG
violates the maximality (2.1) for H. We may assume by symmetry that g1 2 G1 [ G2
belongs to G1. Hence
Gg10 G1; (G0 \ H)g1 G1 \ H0 and |G0 \ H| |G1 \ H0|:
However H \ G0 = H \ G1 by assumption on H, hence |H \ G1| |H0 \ G1|. By
maximality (2.1) with (2.6) follows
|H \ G0| = |H \ G1| = |H0\ G1|: (2*
*.10)
We now consider hg12 H0 with h 2 H subject to (2.2). Such an h 2 H exists a*
*s we
assume that the lemma does not hold. Obviously
hg1= g2. .g.nyg-1n. .g.-12
from (2.8). We get that |hg1| = |h|-2 < |h| has shorter length. Hence hg12 H0\G*
*1\G2
by (2.2) is impossible, so necessarily hg12 G1 [ G2. On the other hand h 62 H \*
* G0 by
(2.2), hence hg12 H0\ (G1 [ G2) is a `new' element when compared with (H \ G0)g1
H0\ (G1 [ G2), so
|H \ G0| = |H \ (G1 [ G2)| < |H0\ (G1 [ G2)|
which contradicts (2.10), and (2.9) follows.
We continue assuming that the lemma does not hold. Now we want to exploit t*
*he
fact (2.9) that H \ G1 is relatively large. Let h 2 H \ G1 \ G2 still be expres*
*sed as in
(2.8) in normal form. If x 2 H \ G1 then also hx 2 H can be represented like h *
*as
hx = g1x. .g.mxyxg-1mx. .g.-11x= gxyxg-1xwith gix2 Z: (2*
*.11)
for some yx 2 G1 [ G2 and gx = g1x . .g.mxwith factors which are representatives
alternating from G1 and G2, respectively.
Hence
g1. .g.nyg-1n. .g.-11x = g1x. .g.mxyxg-1mx. .g.-11x
8
and by uniqueness of factors from the left of the reduced normal forms for tran*
*sversals
follows element-wise g1= g1x; : :;:gn= gmx, hence m = n and g = gx for all x 2 *
*H \ G1.
By (2.11) this is to say that all elements in the coset h(H \ G1) of H \ G1 are*
* conjugate
by the same element g. Accordingly, if
-1
X = H \ (G1 [ G2)g
then h(H \ G1) X H and hence |H \ G1| |X|.
We now consider H0 = Hg and note that Xg H0 \ (G1 [ G2). Cosets have the
same size, hence using (2.6) and (2.1) for H we get
|H \ G1| = |h(H \ G1)| |X| = |Xg| |H0\ (G1[ G2)| |H \ (\G1[ G2)| = |H \ G1|
and equality holds. Hence
h(H \ G1) = H0\ G1 orh(H \ G1) = H0\ G2;
the coset is a subgroup which is only possible if h 2 H \ G1 or h 2 H \ G2, whi*
*ch
however contradicts our choice of h. The lemma holds. 2
Note that HNN extensions are obtained by particular successive free product*
*s with
amalgamation, see [16, p. 182] or [15]. From Lemma 2.1 we have the immediate
Corollary 2.2 Let G be any group, and OE : G0 ! G1 be an isomorphism between *
*two
subgroups of G. Consider the HNN extension G* = .*
* If H
is a finite subgroup of G*, then there exists a y 2 G* such that Hy is containe*
*d in G.
The following lemma describes centralizers of finite subgroups in free prod*
*ucts with
amalgamation.
Lemma 2.3 Let G* = G1 *G0 G2 be the free product of G1 and G2 amalgamating a
common subgroup G0. Let H G1 be a non trivial finite subgroup and let x 2 G* b*
*e an
element which commutes with all elements of H. Then either x 2 G1 or Hg G0 for
some g 2 G*.
Proof. Suppose [x; H] = 1 and x 62 G1. Express x in a reduced normal form
x = g1g01. .g.ng0n;
9
that is, gi 2 G1 \ G0, (1 < i n) and g0i2 G2 \ G0, (1 i < n). The relation
h-1x-1hx = 1 yields the following
h-1g0-1ng-1n. .g.0-11(g-11hg1)g01. .g.ng0n= 1:
By the normal form theorem for free products with amalgamation [15, Theorem 2.6
p. 187], this is only possible if g-11hg1 2 G0 for all h 2 H0. This concludes *
*the proof.
2
By similar arguments we have
Lemma 2.4 Let G be any group, and OE : G0 ! G1 be an isomorphism between two
subgroups of G. Consider the HNN extension G* = .*
* If H
is a non trivial finite subgroup of G* and x 2 G* such that [x; H] = 1, then x *
*2 G.
3 Group theoretic approximations of G
We fix a suitable group H and write bH= Aut(H). Moreover, view H bHas subgroup.
We also fix an uncountable regular cardinal . As usual CG0(G) and NG0(G) denot*
*e,
respectively, the centralizer and the normalizer of a subgroup G in a group G0.
Let pInn (G) denote the set of partial inner automorphisms, which are the i*
*somor-
phisms OE : G1 ! G2 where G1; G2 G such that OE can be extended to an inner
automorphism of G. Hence pInn (G) are all restrictions of conjugations to subg*
*roups
of G. In addition we will use the following
Definition 3.1 Let M N be groups, then x 2 N is nice over M in N if for any
s; t 2 M with x = sxt follows s = t-1.
Obviously nice elements over M in N as in the Definition 3.1 are also nice *
*over M
in G if M N G. This will be used very often in Section 4 and 5. We next consi*
*der
two particular families K* K of groups which will be used to approximate our g*
*roup
G group theoretically. An ordering will follow in the next section. The class o*
*f groups
K* will be dense in K in the sense that for any group G 2 K is the subgroup of *
*some
group G02 K*. Moreover we will show that |G| . @0 = |G0|.
Definition 3.2 1. K consists of all groups G with |G| < such that H bH G,
and any isomorphic copy of H in G has trivial centralizer in G. That is,
K = {G : bH G; |G| < ; ifH ~=H0 G; x 2 G with [H0; x] = 1; then x = 1}:
10
2.K* is the class of all groups G in K such that any isomorphism between H *
*and a
subgroup of G is induced by conjugation with an element in G.
As H is suitable, we have bH 2 K*.
Lemma 3.3 If G and G0 are in K then G * G02 K. Moreover, all elements in G *
*are
nice over G0 in G * G0.
Proof. Suppose that H0 G*G0with H0 ~=H and x 2 G*G0such that [H0; x] = 1. By
Lemma 2.1, we can suppose that (H0)y G. Hence [H0; x] = 1 implies [(H0)y; xy] *
*= 1
and we may assume that H0 G and [x; H0] = 1. If x 62 G, then express x in redu*
*ced
form and consider the commutator h-1x-1hx = 1 for any h 2 H. Replace a first ch*
*oice
h 6= 1 by a different one if the first G-factor of (x)-1 is cancelled by h-1. *
*Hence the
normal form of the commutator shows non-trivial factors and the commutator can *
*not
be 1 by the normal form theorem for free products [15, p. 175]. This is impossi*
*ble, hence
x 2 G 2 K implies x = 1.
The second statement of the lemma follows by an immediate length argument. *
* 2
If G is any group in K and OE : A ! B is an isomorphism between two subgrou*
*ps of G
isomorphic to H, we want that OE is an partially inner automorphism in some ext*
*ension
G G2 2 K. This follows by using HNN extensions as we next show.
Lemma 3.4 Let G 2 K and B G be a subgroup isomorphic to H. Then there is
G G1 2 K such that Aut (B) G1.
Proof. Let bB= Aut (B) and N = NG (B). If bB G then let G1 = G. Suppose that
Bb 6 G. Note that N = G \ bB, so we can consider the free product with amalgama*
*tion
G1 := G *N Bb. We shall show that G1 2 K. Let H0 G1 be a subgroup isomorphic
to H and 1 6= x 2 G1 such that [H0; x] = 1. By Lemma 2.1 we can suppose that
H0 G or H0 bB. Suppose that H0 G, the other case is easier. Let x = g1g2. .g*
*.n
be written in a reduced normal form. If x = g1 2 G then x = 1 since G 2 K, and
this is a contradiction. Hence x = g1 2 Bb \ N. As in Lemma 2.3 we deduce that
H0 = (H0)g1 N, thus H0 = B since B is suitable. Hence g1 2 N is a contradiction*
*. If
n = 2, then we obtain (H0)g1 = (H0)g2 = B = H0. So both g1 and g2 are in N, whi*
*ch
is a contradiction. Similarly, if n 3 we have that g2 and g3 are in N. This is*
* again
impossible. This concludes the proof. 2
11
By the previous lemma we can suppose that if B G 2 K, and if B ~= H, then
Bb G as well. If A; B G, A ~=B ~=H and Ab, bB are conjugate in G then A and B
are also conjugate. Indeed, if g 2 G such that g* : bA-! bB, then Ag bBis a su*
*bgroup
isomorphic to B, hence Ag = B by Definition 1.
Lemma 3.5 Let G 2 K and B bB G. Suppose that H and B are not conjugate in
G. Let OE : bH! bBbe any isomorphism. Then the HNN extension
G1 =
is also in K.
Proof. Let H0 G1 be a subgroup isomorphic to H and 1 6= x 2 G1 such that
[H0; x] = 1. By Corollary 2.2 we can suppose that H0 G already. Let
x = g0t"1g1t"2. .g.n-1t"ngn
be written in a reduced form in G1, where gi 2 G and there is no subword t-1git*
* with
gi2 bH or tgit-1 with gi2 bB(see [15, p. 181]).
If x = g0 2 G then x = 1, since G 2 K. This yields a contradiction. Thus n *
* 1.
We have [h; x] = h-1x-1hx = 1 for every h 2 H0. In other words, for a fixed h 6*
*= 1, the
following holds
h-1g-1nt-"n . .t.-"1(g-10hg0)t"1. .t."ngn = 1: (*
*3.1)
By the normal form theorem for HNN extensions ([15, p. 182]), either "1 = 1 and
g-10hg0 2 Hb, or "1 = -1 and g-10hg0 2 Bb. Suppose that "1 = 1, the other case*
* is
analogous. Then (H0)g0 Hb, thus (H0)g0 = H from `suitable', and we can replac*
*e in
(3.1) the subword t-1(g-10hg0)t by OE(g-10hg0) 2 B. Repeating the same argumen*
*t we
obtain that "i= 1 or -1. Hence one of the two possibilities holds depending on *
*"2 = 1
or "2 = -1. We have either
g*0 OE g*1 OE g*n
id : H1 -! bH- ! bB-! bH -! . .-.! H1
or
g*0 OE g*1 OE-1 g*n
id: H1 -! bH- ! bB-! bB- ! . . .-! H1:
OE g*1
In the first case we have an isomorphism g*1OE : bH -! bB- ! bH. Since Hb is *
*complete
there is g 2 bHsuch that g*1OE = g*. This yields OE = (g-11g)*, i.e. Hb and bBa*
*re conjugate,
12
and thus H and B are conjugate, but this is impossible by hypothesis. In the s*
*econd
case we have g1 2 bBby completeness. But, on the other hand g1 62 bBsince x is *
*written
in a reduced form. We conclude that G1 is in K. 2
Lemma 3.6 Let A bA G 2 K and B bB G. If OE : A -! B is any isomorphism,
then there is G G2 2 K such that OE 2 pInn (G2). Moreover, G2 can be obtained *
*from
G by at most two successive HNN extensions.
Proof. If OE 2 pInn (G) we take G2 = G. Suppose that OE 62 pInn (G). If bH and *
*bAare
conjugate, we take G1 = G. Otherwise, we consider the HNN extension
G1 =
where OE1 is any isomorphism between bH and bA. By Lemma 3.5 we know that G1 2 *
*K.
Now, if H and B are conjugate in G1, we take G2 = G1. It follows automatically *
*that
OE 2 pInn (G1), since bH is complete. If bH and bBare not conjugate in G1, we c*
*onsider a
new HNN extension
G2 =
where OE2 is any isomorphism between bH and bB. Again G2 2 K by Lemma 3.5. In t*
*hat
case we have an isomorphism (t-12)*OEt1* : bH ! bH, which equals g* for some g *
*2 bH by
completeness. Thus OE = (t2gt-11)* A. This shows that OE 2 pInn (G2). 2
Lemma 3.7 Let G 2 K and suppose that G02 K or G0 does not contain any subgr*
*oup
isomorphic to H. Let g 2 G and g02 G0with o(g) = o(g0). Then (G * G0)=N 2 K whe*
*re
N is the normal subgroup of G * G0 generated by g-1g02 G * G0.
__
Proof. The group G = (G * G0)=N is a free product with amalgamation, hence G and
__ *
* __
G0can be seen as subgroups of G respectively. Suppose that we have a subgroup H*
*0 G
__
isomorphic to H and x 2 G such that [H0; x] = 1. By Lemma 2.1 we can assume that
H0 is already contained in G. Suppose that x 6= 1. By Lemma 2.3 it follows that*
* either
x 2 G or a conjugate of H0 is contained in . In the first case x = 1 from G *
*2 K is a
__
contradiction. The second case is obviously impossible. Thus G 2 K. 2
The proof of the next lemma is obvious.
13
Lemma 3.8 (a) Let fl < and {Gi : i < fl} be an ascending continuous chain*
* of
S
groups in K. Then the union G = Gi also belongs to K.
i.
We need that N = G. There are two natural cases depending on the order of x. The
case that x has infinite order is taken care by the next Proposition 3.10. Henc*
*e assuming
that all elements of infinite order are conjugate, a consequence of Proposition*
* 3.10, we
only need to note that any element g of finite order can be written as a produc*
*t of two
elements of infinite order, just take y from a different factor then g = (gy)y-*
*1. Hence
G = N. If x has finite order, then there is a conjugate y of x such that xy has*
* infinite
order. Hence xy 2 N and the first case applies.
Proposition 3.10 Let G be a group in K. Let g; f 2 G, where o(f) = o(g) = 1 *
*and
*
*__
g does not belong to the normal subgroup generated by f. Then there is a group *
*G 2 K
__ __
such that G G and g is conjugate to f in G .
14
Proof. Let ff : ! be the isomorphism mapping f to g. By hypothesis
__
ff 62 pInn G. As in Lemma 3.6 consider the HNN extension G = *
*. We
__ __
must show that G 2 K. Clearly |G | < and consider any H0 with H ~= H0 G and
__ __
any x 2 G with [H0; x] = 1. As above we may assume that H0 G and x 2 G with
[H0; x] = 1. Now we apply Lemma 2.3. 2
The last lemma of this section is not needed for proving the Main Theorem b*
*ut it
is used to show the additional property (e) of the group G in Section 1 mention*
*ed after
Main Theorem.
__ __
Lemma 3.11 If g 2 G 2 K, then there is a group G 2 K, such that G G , with
___ __
|G | = |G| . @0 and g 2 H(G ).
Proof. Suppose that o(g) = 1 and that g 62 H(G). Let H1 and H2 be two isomorphic
copies of H. Choose a non trivial element h 2 H and let h1 and h2 be its copies*
* in H1
and H2 respectively. Now define
__
G = (G * H1 * H2)=N
__
where N is the normal subgroup generated by g-1h1h2. Then G 2 K by Lemma 3.7 and
__
moreover g 2 H(G ).
If o(g) = n < 1 we first embed G (G * K)=N where K =
and N is the normal closure of g-1x1x2. Then by the Lemma 3.7 (G * K)=N 2 K. No*
*w,
since o(x1) = o(x2) = 1, we can apply the first case. 2
4 Approximations of G as a + -uniform poset
We recall some basic notions of set theory from [12]. In particular cf(ff) den*
*otes the
cofinality of an ordinal infinite cardinal ff and a cardinal is regular if cf(*
*) = .
Throughout let be a fixed uncountable regular cardinal and + it successor. We *
*will
write P< () for the set of all subsets of of cardinality < . From GCH we have
|P< ()| = < = .
In this section we have to fit the group extensions of the last sections in*
*to a poset
P defined in the appendix A of the paper. Let and H be as in Main Theorem, and
H bH as before. Write
f+ := {(ff; i) : ff 2 + ; i < }: (*
*4.1)
15
with the lexicographical ordering. Hence f+ is a well-ordered set of cardinalit*
*y + , an
ordinal < ++ . Hence
ss : + -! f+ (ff -! (ff; 0)) is a canonical embedding.
If x = (ff; i) 2 f+ with i < , we write ||x|| = ff and call ff the norm of x. W*
*e define
the domain of a subset X of f+ as the set dom X = {||x|| : x 2 X}.
The following picture illustrates how we can embed for instance H * Hffin f*
*+, with
dom (H * Hff) = {0; ff}.
0 (1; 0). . . (ff; 0) (ff + 1; i). . .
__________________________________________________________________|f+|||_*
*||_||
||H i i ii Hff\ {1} |
| i i ||| .*
* ||
| i i |
|? i i |
__________________________________________________________________ss|?ii)*
*||||
01 . . . ffff + 1 . . . +
Definition 4.1 Let u + be a subset of cardinality < with 0 2 u. Then a grou*
*p G
of size |G| < is called an u-group if the following holds:
(a) Dom G f+ and dom G = u, where Dom G denotes the underlying set of elem*
*ents
of the group G. We will identify Dom G = G.
(b) For every 0 6= ffi 2 + the subset G \ ss(ffi) is a subgroup of G. Moreove*
*r G \ ss(ffi)
belongs to K* given in Section 3.
We will rewrite the elements p = (ff; u) 2 P (see the Appendix A) in the fo*
*rm
p = (Gp; up) or simply p = (G; u) where G is a u-group. If u is fixed, then G *
*runs
through all u-groups of cardinality < . By GCH this is a set of cardinality , *
*hence
this modification of P agrees with the requirement that (only) ff < (codes the*
*se
algebraic structures). Next we define an ordering on P which will use `nice' e*
*lements
from Definition 3.1. Now we say that
p q in P, which is the case if and only if the following two conditions hold:
1.Gp Gq
2.If ffi 2 + and x 2 Gp is nice over Gp \ ss(ffi) in Gp, then x 2 Gq is nic*
*e over
Gq \ ss(ffi) in Gq.
16
Theorem 4.2 (P; ) is a + -uniform partially ordered set.
Proof. We must define elements in P and have to check the conditions listed in
Definition A.1:
First we trade Hb into a 0-set for example, as indicated by the diagram. H*
*ence
(Hb; {0}) is our first element in P.
Suppose Gi is a ui-group for each i = 0; 1; 2 such that G0 = G1 \ G2. We tu*
*rn the
free product G* = G1 *G0 G2 into a `weak' u-group for u = u1 [ u2 which is a u-*
*group
except that the subgroups G* \ ss(ffi) (ffi < + ) may not be in K*.
If g 2 G* = G1*G0 G2\ G1\ G2 then choose a transversal of G* as in (2.7) an*
*d write
g uniquely as indicated there. Turn g into an unused ordinal g 2 f+ such that i*
*ts norm
||g|| is the maximum of the norms of those factors. With HNN extension we can d*
*eal
similarly, which is left as an exercise. In order to satisfy (b) of the Defini*
*tion 4.1, we
apply Lemma 3.8 extending G*\ ss(ffi) accordingly and identifying with unused o*
*rdinals
of the intervals related to u. We see that G* becomes a subgroup of the set f+*
* with
dom G* = u and G* is a subgroup of some u-group G obtained by iterated applica*
*tions
of free products with amalgamation and unions of such chains. Hence it follows*
* from
Definition 3.1 that nice elements in G0 over G0 \ ss(ffi) remain nice in G \ ss*
*(ffi), which
we will use silently to check 1; : :;:8 in Definition A.1:
1. Let p; q 2 P such that p q and p = (Gp; up) and q = (Gq; uq). Then Gp *
*Gq
and dom Gp dom Gq, or equivalently dom p dom q.
2. Let p; q; r 2 P such that p; q r. With the same notation as above we ha*
*ve that
Gp, Gq are subgroups of Gr and all of them belong to K*. Consider G0 Gr generat*
*ed
by Gp and Gq. It is clear that u = dom G0= dom p [ dom q. Using the fact that G*
*r is
in K*, we can add to G0 the elements of Gr, with norm in u and obtain a new gro*
*up
G002 K* such that G0 G00 Gr. Hence r0= (G00; u) is the required element in P.
Condition 4. can be shown similarly, 3., 5. and 6. are obvious in view of t*
*he previous
remarks.
7. (Indiscernibility) Suppose that p = (Gp; up) 2 P and ' : up ! u0 is an *
*order-
isomorphism in + . We define a set
G0= {('(ff); i) 2 f+ : ff 2 up; (ff; i) 2 G} f+
and give to G0multiplication canonically induced by G. The map ' induces a map *
*of p
to some '(p) = (G0; u0) 2 P which we also denote by '; it is order preserving o*
*n up ! u0
17
and a group isomorphism on G ! G0. The fact that q p implies '(q) '(p) is also
clear.
8. (Amalgamation property) Using the same notations let be Gp and Gq from p
and q respectively and let r = (Gr; ur) be such that Gr = Gp \ Gq. The free pro*
*duct
G* = Gp *Gr Gq with amalgamated subgroup Gr by Lemma 3.9 belongs to K . By
Lemma 3.8 there is a group G 2 K* such that G* G. Using the remarks at the
beginning we can trade G into an (isomorphic) u-group which we also call G with
dom G = u = dom Gp [ dom Gq. Hence s = (G; u) is as required for 8. 2
It may help to make the following
Definition 4.3 Let Gi be ui-groups for i = 1; 2. A map : G1 ! G2 is a stro*
*ng
isomorphism if : G1 ! G2 is a group isomorphism which preserves the order on
dom Gi, this is to say that
dom G1 ! dom G2 ( ||x|| ! || (x)|| ) (x 2 G1)
is an order isomorphism.
The map defined in 7. is such a strong isomorphism.
Below we will introduce certain density systems on P which will ensure the *
*require-
ments stated in the Main Theorem.
The group G at the end will be
[
G = Gff; (*
*4.2)
ff<+
where each Gffis the union of all groups in the directed system Gffover Pff( se*
*e Definition
A.2 ), where
Pff= {p 2 P : dom p ff}:
Every Gffhas cardinality , so that (4.2) is a + -filtration of G. For the rest*
* of this
section we fix the following notation: Let be ff < fi < + , u v + with |v| < *
*, and
define
E := {p 2 P=Gff: v dom p v [ fi}:
Recall from Definition A.2 that p 2 E if and only if p ff 2 Gff, in our settin*
*g p =
(Gp; up) this is to say that (Gp\ss(ff); up\ff) 2 Gff. Note that this will foll*
*ow for density
systems (below) from condition 2 of the ordering on P. We define
18
the density system for |G| = + to be the set
Dff(u; v) := {p 2 E : u [ {ff} dom p v [ fi}: (*
*4.3)
Proposition 4.4 The collection Dffof Dff(u; v) as in (4.3) is a density syste*
*m over
Gff.
Proof. We will use the notation above and from Definition A.2. It is clear th*
*at
Dff(u; v) is closed upwards in E. To show that Dff(u; v) is dense in E it is e*
*nough to
consider the case that ff 62 u and q = (G; u) 2 E. As in the proof of Theorem *
*4.2 we
can find a group G02 K*, such that G G0, which contains a subgroup H0 isomorph*
*ic
to H and ff 2 dom H0 v [ fi. In particular, we have ff 2 dom G0. Moreover,*
* the
group G0can be constructed in such a way that dom G0 v [ fi. By the remarks abo*
*ve
p := (G0; dom G0) 2 E, nice elements in G are also nice in G0, hence q p 2 Dff*
*(u; v)
shows density. The second condition in Definition A.3 can be verified similarly*
* to the
proof of 7. in Theorem 4.2. 2
The density systems to make G simple
If x; y 2 Gffand o(x) = o(y) = 1 then let
p
Dx;y(u; v) = {p 2 E : p = (Gp; up) ; u up v [ fi; x 2 }; (*
*4.4)
p z p
where = .
Proposition 4.5 Dx;yas in (4.4) is a density system over Gff.
q
Proof. If q = (Gq; uq) 2 E we may assume that y; x 2 Gq. If x 62 we can
q
construct a v-group G represented by p 2 E extending q such that x 2 : Ap*
*ply
Proposition 3.10 and define G1 = , as an HNN extension, wher*
*e t
is a new element in v with ||t|| = ||x||. Using again the argument from the pr*
*oof of
Theorem 4.2 we can also find G1 G with G 2 K* and dom (G) v \ fi which gives
p q = (G; u) 2 E and x and y are conjugate in G, hence q 2 Dx;yshowing density.
2
The density systems to trade monomorphisms of G into inner automorphism
For any subgroup K Gffof cardinality < and any monomorphism : Gff! Gff
define the set D K (u; v) as
{p = (Gp; up) 2 E : K Gp 9y 2 Gp with K = y* K; dom p v [ fi} (*
*4.5)
19
Note that the just defined system (running over all and K) of sets D K (u*
*; v) has
(only) size .
We also have a special case of the last density system (4.5) which we state*
* explicitly
because it serves for a different purpose.
The density system to conjugate copies of H in G
In this case we choose for each H0 Gffwith isomorphism : H ! H0 the set
D (u; v) = {p = (Gp; up) 2 E : H0 Gp and 9g 2 Gp; H = g* H} (*
*4.6)
Proposition 4.6 The collection of all D K (u; v) as in (4.5) [respectively D *
*(u; v) as
in (4.6)] is a density system over Gff.
Proof. Apply Lemma 3.8 and Definition 3.2: If p 2 E then we find a group G such
that K = y* K for some y 2 G by HNN extension. By Theorem 4.2 we also find
G G02 K* and we trade G0into a v-group with ||y|| = ff and v = u [ {ff} as we *
*did
before. Hence Proposition 4.6 follows. 2
The density for many nice elements
For ff as above we also choose
D(u; v) = {p = (Gp; up) 2 E : 9 q 2 E; 9x 2 Gp torsion-free and * Gq = Gp*
*}(4.7)
The density can easily be checked as before.
5 Proof of the Main Theorem
We will fix for the rest of this paper a particular word of a free group (i.e. *
* a term in
group theory) o(x1; x2; x3; x4) = [x1x2; x3x4] which is the commutator of produ*
*cts in
free variables x1; : :;:x4. We will also use the notion of a group isomorphism *
*which is
at the same time `level preserving', the strong isomorphism from Definition 4.3
The main result for proving the Main Theorem is the following lemma concern*
*ing
this word.
Main Lemma 5.1 Let P be the + -uniform poset defined in the last section. *
*Assume
that the following three properties hold in P.
1. There are ordinals ffi1 < ffi2 < ffi3 < ffi4 and approximations pi 2 Pffi*
*i+1(i = 1; 2; 3)
and p4 2 P with pi ffii= p0 2 P for i = 1; : :;:4.
20
*
* p0
2. There is a nice element x1 2 Gp1\ Gp0 over Gp0 and an element y1 2 Gp1\ x*
*G1 .
3. Let 'i : Gp1 ! Gpi for i = 2; 3; 4 be a strong isomorphism and 'i(x1) = x*
*i,
'i(y1) = yi in Gpirespectively.
Then we can find in P an approximation q p1; : :;:q4 such that in the gr*
*oup Gq
we have
o(x1; x2; x3; x4) = 1 buto(y1; y2; y3; y4) 6= 1:
Proof. We assume the hypothesis of the lemma, in particular we have elemen*
*ts
xi; yi with (i = 1; 2; 3; 4) in the appropriate groups Gpi. From P we can choo*
*se an
approximation q = (G; v) p1; p2 such that G = Gp1*Gp0 Gp2, hence x =: x1x2 2 G.
First we want to show that
1.For any word oe(x; t) = (x1x2)n1t1(x1x2)n2t2. . .= 1 for a finite set t= *
*{t1; : :;:tk}
from Gp0 follows that the ti's commute with x.
2. x is nice inG over Gp0:
At the end we want to apply the normal form theorem for free products with *
*amal-
gamation [15] to the equation
1 = (x1x2)n1t1(x1x2)n2t2. . .withti2 Gp0 and ni2 Z \ 0:
Naturally we can distinguish four cases, where we use the following notation: L*
*et x"11be
the last xi-term which appears in (x1x2)n1, similarly let x"22be the first xi-*
*term which
appears in (x1x2)n2.
Obviously we have the following possibilities:
1.n1 > 0 ) (1 = 2 and "1 = 1)
2.n1 < 0 ) (1 = 1 and "1 = -1)
3.n2 > 0 ) (2 = 1 and "2 = 1)
4.n2 < 0 ) (2 = 2 and "2 = -1)
Hence the displayed equality can only hold if 1 = 2 2 {1; 2}. To be definit*
*e we take
1 = 2 = 1 which by the cases implies "1 = -1; "2 = 1; n1 < 0 and n2 > 0. Hence *
*the
term in the last displayed equation connecting xn1 and xn2 is x-11t1x1 and the *
*equality
21
-1
for normal forms forces s =: x1 t1x1 2 Gp0. Hence x1 = t1x1s-1 which by hypothe*
*sis is
a nice element, hence s = t1 and [x1; t1] = 1. Using '2 also [x2; t1] = 1 and i*
*nduction
shows that all the ti's commute with x1 and x2 and in particular any equation o*
*f the
form txs = x with s; t 2 Gp0 implies s = t-1, hence x is nice.
Next we consider the relationship of the xi's and the yi's. We have the fol*
*lowing
Claim 5.2 If there are group terms (words) y1 = oe1(x1; t) 2 Gp1 and y1y2 = o*
*e2(x; t0)
with t; t0 Gp0, then there is an s 2 Gp0 such that y1 = oe1(x1; t) = s-1x1s:
Proof. Using the action of '2 we have
oe1(x1; t)oe1(x2; t) = y1y2 = oe2(x1x2; t0):
By hypothesis yi2 Gpi\ Gp0, hence the displayed element has length 2. Let oe2(x*
*; t0) =
t1xn1t2: :t:kxnk be in canonical form as before. Hence the normal form forces k*
* = 1 and
if |n1| > 1 then t1(x1x2)n1t2 has length > 2, so also n1 2 {1}. We arrive at tw*
*o cases
oe1(x1; t)oe1(x2; t) = t1x1x2t2 (*
*5.1)
or
oe1(x1; t)oe1(x2; t) = t1x-12x-11t2: (*
*5.2)
In the first case normal form forces that there is s 2 Gp0 such that oe1(x1; t)*
* = t1x1s
as well as oe1(x2; t) = s-1x2t2. Application of '2 also gives oe1(x2; t) = t1x*
*2s, hence
t1x2s = s-1x2t2. Recall that x2 (like x1) is nice, hence t1 = s-1 and s = t2, t*
*he claim
follows (in this case). In the other case we have an element (5.2) which is wr*
*itten in
normal form at the same time as products from exchanged factors Gp1 and Gp2 whi*
*ch
is impossible. 2
In order to complete the proof of the Main Lemma 5.1 we define two more ext*
*ensions
of Gq in P. Let q02 P be given by dom q0= dom p3[dom p4 such that as groups we*
* have
0 p p q p p
Gq = G 3*Gp0G 4. Recall that G = G 1*Gp0G 2. Hence we find a strong isomorphism
(which is order preserving on dom : :):which is
0 -1
' : Gq ! Gq extending '3; '4'0 ;
hence
'(x1) = x3; '(x2) = x4; '(x) = '(x1x2) = x3x4; '(y1) = y3; '(y2) = y4
22
and we let
x1 = x = x1x2; x2 = x3x4; y1 = y1y2; y2 = y3y4
0 p *
* 1
which are in the appropriate factors Gq respectively Gq but not in G 0, moreove*
*r '(x ) =
x2; '(y1) = y2. If there is a group term y1 = oe(x1; t) with t Gp0 then we can*
* apply
p0
the last Claim 5.2 to see that y1 = s-1x1s 2 (x1)G which was excluded by hypo*
*thesis
of the Main lemma. We conclude that
0
y1 62 Gq and similarlyy2 62 Gq : (*
*5.3)
Now we define a final approximation r 2 P with dom r = dom q0 [ dom q. Group
theoretically we get Gr in several steps:
To easy notation let
0 0 1 0 2
K0 = Gp0; K1 = Gq; K2 = Gq ; K1 = K1; K2 = K2;
0 0
and define L0 = K01*K0 K02= , and if N = <[x1; x2]L > / L is the n*
*ormal
subgroup of L0 generated by the commutator [x1; x2] then let L = L0=N. However
K0 \ N = 1 and K0 L canonically, hence L is the free product of K0 with the fr*
*ee
abelian group of rank 2. Using only the group operation of Section 4*
*, the
group L now obviously can be made into an element in P. Note that also N \K0i= *
*1, we
get a canonical embedding K0i L and can consider Mi = Ki*K0iL for i = 1; 2; fin*
*ally
put Gr = M1 *L M2. From the normal subgroup N follows in Gr that [x1; x2] = 1. *
*On
the other hand from (5.3) it follows that yi 62 K0i, yi 2 Gpihence y1 = y1y2 2 *
*Gq \ K01
0 0 i 0 1 2
and similarly y2 2 Gq \ K2. By definition of Mi also y 2 Mi\ Ki and [y ; y ] c*
*an not
cancel in Gr, this is to say that [y1; y2] 6= 1.
Using now directly the Main Theorem 1.11 (which is in terms of model theory*
*) from
the forthcoming book Shelah [21] (or slight modifications in [11, 20] or [19] w*
*e get the
following proposition. Its proof like earlier `black boxes' (see the appendix *
*of [6] for
instance), also this case is based counting arguments but using 3 on {ff 2 + : *
*cf(ff) =
} and {ff 2 + : cf(ff) = !} as well. The latter explains why the generalized co*
*ntinuum
hypothesis gets into the proposition.
The statement of the proposition depends on P, the density systems construc*
*ted in
Section 4 and also (substantially) on the Main Lemma 5.1.
Proposition 5.3 1. Assuming ZFC + GCH, there is an ascending sequence of o*
*r-
dinals iff< + and a continuous ascending chain of admissible ideals Gff P*
*iff
which meets all density systems constructed in Section 4.
23
S S
2. Let Gff= Gffand G = ff<+Gff:
3. If for all ff < + there is a nice element xff2 G \ Gffover Gffand there i*
*s yff2 G \
xGffff, then there are four ordinals ff1 < ff2 < ff3 < ff4 such that [xff*
*1xff2; xff3xff4] = 1
and [yff1yff2; yff3yff4] 6= 1.
For simplicity we will say that iff= ff without less of generality. We will ap*
*ply this
black box for proving the Main Theorem 1.
Proof. Obviously H G because any approximation has H as subgroup.
G has cardinality + : By Proposition 5.3 we have that for every ff < + , there
is a group G in G such that ff 2 dom G. Hence dom G = + and + = |G| follows
immediately.
Property (d): Let x; y 2 G be both of infinite order. We can apply Proposition*
* 4.5
and the density condition from Proposition 5.3 to see that x and y are conjugat*
*e in G.
If x has finite order and y has infinite order we can easily find an element x0*
*= xz 2 G
of infinite order such that z-1y has infinite order as well. Hence the first c*
*ase applies
and similarly we work of also y has finite order. Hence in any case x 2 .
Property (a) follows from (4.6) and Proposition 4.6.
Property (b) was carried on inductively by our choice of K*, hence this also ho*
*lds for
G. Finally we have to show that
Property (c) holds, this is to say that
any monomorphism : G ! G is an inner automorphism of G.
Suppose : G ! G is a counterexample, a monomorphism which is not inner. We fi*
*rst
want to show that the following holds.
There isff* < + such that (x ) 2 x Gfffor allx 2 G \ Gff*: (*
*5.4)
Otherwise
for allff < + there isxff2 G \ Gffwith yff= (xff) 62 xGffff (*
*5.5)
>From (4.7) it is easy to see by a change of elements xffthat
we may also assume that each xffis nice overGffin G: (*
*5.6)
By Proposition 5.3, based also on the Main Lemma 5.1 together with (5.5) we*
* can find
four ordinals ff1 < ff2 < ff3 < ff4 such that [xff1xff2; xff3xff4] = 1 and [yff*
*1yff2; yff3yff4] 6= 1.
24
However 1 = ([xff1xff2; xff3xff4]) = [yff1yff2; yff3yff4] 6= 1 is a contradict*
*ion. Hence we
may assume that (5.4) holds. 2
First we claim that
8X G; |X| = + ) (9 X0 X; 9y* 2 G; |X0| = + ; (x) = xy* 8x 2 X0): (*
*5.7)
Let # z = min {ff < + : z 2 Gff} for any z 2 G. Let S be the set of all limit o*
*rdinals
ff < + with ff* < ff. Hence S is a stationary subset of + of cardinality + and *
*from
(5.4) we can choose a sequence xff2 X \ Gffand yff= (xff) 2 xGfffffor all ff 2*
* S.
We may assume that {# yff: ff 2 S} is bounded in + . For otherwise the function
f : (S ! + )(ff ! f(ff) = # yff) satisfies f(ff) = # yff< ff and is regressive.*
* By Fodor's
lemma there is a sationary subset S0 of S and fi < + such that f(ff) = fi for a*
*ll ff 2 S0,
see Jech [12, p. 59, Theorem 22]. As |S0| = + we can replace S by S0. We now co*
*ntinue
using S and choose the new X = {xff: ff 2 S}. From |Gfi| = < |X| = + we also f*
*ind
an equipotent subset of X, call it X again, and
y* 2 Gfisuch that (x) = xy* for allx 2 X:
The claim (5.7) is shown.
We now choose a set X = {x1ff2 G \ Gff: ff < + } of nice elements over Gffi*
*n G with
|X| = + and apply (5.7). Hence we may assume that X = y** X for some y* 2 G.
Replacing by (y-1*)* we may assume that y* = 1, hence
X = idX:
We now assume for contradiction that 6= idG: There is an x* 2 G such that (x*
**) 6= x*.
Next we choose a second sequence of nice elements over Ga which is {x2ff2 G \ G*
*ff:
(x2ff) 6= x2ff: ff < + }. If a first choice fails for any subsequence, the we *
*multiply each
of these elements by x*. For ff > ff* the new elements are obviously nice and d*
*o what
we want. We apply once more the claim (5.4) and find some ff* and tff2 Gff*such*
* that
(x2ff) = (x2ff)tfffor all ff* < ff < + . By a pigeon hole argument we may assu*
*me that
t = tfffor all ff* < ff < + . Hence we found a sequence of pairs of nice elemen*
*ts over
Gff*:
x1ff; x2ff2 G \ Gffwith y1ff= (x1ff) = x1ff; y2ff= (x2ff) = (x2ff)*
*t 6= x2ff:
Recall the properties of P: If ff* < ff < ff1 < ff2 we can choose p1 2 Pff1suc*
*h that
xiff1; yiff12 Gp1\ ss(ff); i = 1; 2 and t 2 Gp1\ ss(ff*). Moreover we find p2 2*
* Pff2such that
25
Gp1\ Gp2 = Gp1\ ss(ff) and let p0 = p1 ff = (Gp1\ ss(ff); dom (Gp1\ ss(ff))): *
*There is
a `level preserving' strong isomorphism
' : Gp1! Gp2 with' Gp0= idGp0;
which carries the xiff1; yiff1's to Gp2. Let '(xiff1) = xiff2which is another *
*nice element
over Gff*and '(yiff1) = yiff2for i = 1; 2. From t 2 Gp0; ' Gp0 = idGp0 and the*
* above
equations we have x1ff2= y1ff2and (x2ff2)t = y2ff2. We now choose p3 2 P with p*
*1; p2 p3,
hence x1ff1; x2ff2are nice over Gp0. From t 2 Gp0 and the last equalities follo*
*ws that
y1ff1y2ff2= (x1ff1x2ff2) = (x1ff1) (x2ff2) = x1ff1(x2ff2)t 62 (x1*
*ff1x2ff2)Gff*:
On the other hand x1ff1; x2ff22 G \ Gff*, hence
y1ff1y2ff2= (x1ff1x2ff2) 2 (x1ff1x2ff2)Gff*
by (5.4) is a contradiction. Hence = idG is an inner automorphism, which cont*
*radicts
our initial hypothesis and property (c) of the Main Theorem follows.
A The appendix: A standard + -uniform set
Definition A.1 A standard + -uniform partial order is a partial order define*
*d on a
subset P of x P< (+ ). Its elements are pairs p = (ff; u) 2 x P< (+ ). We w*
*rite
u = dom p and call u the domain of p. [The ordinal ff is a code for some alge*
*braic
structures under investigation that u can support, in our case ff represents a *
*group on
the set u.] These approximations p 2 P satisfy the following conditions:
1.(Compatibility of the orders) If p q then dom p dom q.
2.For all p; q; r 2 P with p; q r there is r0 2 P such that p; q r0 r and
dom r0= dom p [ dom q:
3.If {pff: ff < ffi} is an increasing sequence in P of length ffi < then i*
*t has a least
S S
upper bound q 2 P with dom q = dom pff; we say that q = pff.
ff