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STABILITY OF

.N'o-CATEGORICITY

AND

GROUPS

NONABELIAN Ulrich FELGNER

Tubingen, West Germany

Ho -categorical if, up to power ~ No such that G and

A group G is called only one group H of

equivalent. A well-known theorem of Engeler, nonius states that for any group

the

G

isomorphism, there is H

are elementarily

Ryl~Nardzewski

following

and Sve-

properties

are

equivalent:

(i)

G is No-categorical;

(ii)

for every

n:

Th (G)

(iii)

for every

n:

every n-type in

(c f , J.Shoenfield

has only finitely many n-types;

[21] p.91). Th(G)

=

Th(C)

is principal.

{~; GFcI>}

is

the

first-order

theory of c. It is also possible to give a purely algebraic characterization of the concept of N -categoricity. Let G be a group and o x1"",x n and Yl""'Yn be elements of C • The ordered n-tuples (x1 ... ,xn) and (Yl .... Yn) are called au t omo rph i a a l l u e q u i v a l en t: if there exists an aut omorph i sm 11 of C such that IT (x j) = Yj for all

l:Sj-:::;;n. The set a(xl, .. ,x n) of all orderedn-tuples (yl' .... yn) which are automorphically equivalent to (xl" .. ,x is called an n) n-automorphism type of C. It is known that G is No-categorical iff (iv)

for every

n: C' has only finitely many n-automorphism types.

(for C countable) In this paper we try to shed some light

on

the structure of

No-

categorical non-abelian groups. The structure of the No-categorical abelian groups is well-known: it is a result of J.G.Rosenstein [20J tha t an abel ian group Gis No -categorical iff G has a finite expo-

3n T/gEC :ng=O). The classifications of all H - c a t e go 1 rical abelian groups and of all w-stable abelian groups are due to A

nent (i.e.

Macintyre [16].

In the case of non-abelian groups only sporadic re-

sults are known (cf. J.Baldwin-J.Saxl

301

[Z],

Ch.Berline [~],U.Felgner

ULRICH FELGNER

302

[1], J.G.Rosenstein [20], [t1], [22] and G.Sabbagh [2'5J ,[2it]).However, W.Baur, G.Cherlin and A.Macintyre

[3]

recently obtained a com-

plete classification of all totally categorical groups, i.e., groups

No- ca tegorical

which are both

oN, - ca tegorical.

and

One of their re-

sults is that such groups are 'abelian-by-finite'. In this paper we shall consider the more general situation of groups which are No-categorical and A-stable. We have the following main results: ( 1 )

A

No

stable

e

q

o

r - i

o

l: group has finite

a

Lo

eiau-r

l

enq

t

h

,

.No - categorical group has a maximum normal nilpotent sub-

(2) A stable

group. Hence the Fitting subgroup exists and coincides with the Baer Nil-radical and the Hirsch-Plotkin-radical. Let

(3)

G be a stable No - categorical

group.

Then

finite chain of characteristic,definable subgroups {1}

=Ho

and for a l-l:

~

j

~

=

there Hi

exists a

such that

H1 ~ H2 ~ . . . . ~ Hn_ 1 -::;] Hn G n: Hj+1 /Hj is either abelian or a direct sum of fi-

ni te ly many simp l:e No - oa t e qo r-i oa l: groups.

I would like to thank R.Baer for some stimulating conversations. § 1.

UN I FORMLY LOCALLY FIN ITE GROUPS

We shall use the following notation. For any set X, Ca r d i X} denotes the cardinality of ways written multiplicatively. If e is the centralizer of X and

Ne(X)

in e, thus

X

.Groups are al-

is a group and xC;; G then Ce(X) CG(X) = {xEe; 'f/yEX:xy=yx},

is the normalizer of X in G, thus

<

NG(X) ={y EG; yX = Xy}.

Furthermore, X> denotes the subgroup of G genera ted by X. In contrast to this notation, {x; ~ (.x)} is the set of all x which have the property

~.

If X

is a singleton, X

= {x},

then we shall write

{x}

instead of ({x}). If g EG then the conjugacy class of g in G is defined to be gG={hEG ;3xEG: h= x- 1gx}. Hence (gG) is the least normal subgroup of G containing g. Z (G) always denotes the center of the group G. thus H~

e

indicates that

H

is a normal subgroup of

Z (G) = Ce(G).

G.

w = {O, 1,2,3, ... } is the set of all positive integers. A B

set of all functions from noted by -C thus ~ =

l?o

The groups = Th(H).

e

A

into

B.

The cardinality of

Ww

is

the

is

de-

and H are called elementarily equivalent iff Th(G)

We expr e s s this relationship between G and H as

usual

by

STABILITY AND We write H=1,G when H

G'= H.

303

~o-CATEGORICITY

is an elementary substructure of

G.

DEFINITION. A group G is called uniformly-loeaHy-finite i f there is a function r« Ww such that Card (X»):'5 f(Card(X)) for every fini t e subset X of

G.

Our interest in uniformly-locally finite groups is motivated by the following observation: let G be a locally-finite group. Then G is uniformly-locally-finite iff all groups, which are elementarily equivalent to G , are locally-finite. An easy consequence of the Engeler-Ryll-Nardzewski-Svenonius theorem is, that groups are uniformly-locally finite! LEMMA 1. (i)

Subgroups and

homomorphio images

No -

categorical

of un i f oi-ml-u l oo a l-l-gr

finite groups are uniformly-ZooalZY-finite. N ~ G and b o t h , Nand

(ii) If then

G

G/N

J

are uniformly-ZoeaHy-finit(',

is uniformly ZoeaZly finite.

Proof. (i) is obvious. Consider (ii): let f E

Ww

and g E

Ww

be such

that V'x£NLCard(X):::n =+Card«X»~f(n)1 and \;fYS;G/NlCard(Y)~ n =;>Card(Y»)==g(n)]. Define h(n)=(n+g(n))'f((n+g(n))2). The argument given by Kegel and Wehrfritz [1~J,p.2,in the proof of their lemma 1.A.2 can be used to show that Card(X)~

n

implies

Car d ! (X»

for any finite subset

X

of

G,

::::; h t n ) , Whence G is uniformly-locally

finite, Q.E.D. The exponent of a group G is the least positive integer n~l such n that x = 1 for all XEG, if such an n exists; otherwise it is 00. Note that, if the exponent of a group G is 2,3,4 or 6, then G is uniformly-Iocally-finite. This follows

readily from M. Hall's and I.N.

Sanov's solution of Burnside's problem for n= 2,3,4,6(cf.M.Hall pp. 320-338, Magnus-Karras-Solitar [17] pp.379-388). LEMMA 2. A looaZZy-finite group of exponent

a prime

p

[9]

4,6,10,12, 15,20,30

01'

is uniformZy-Zocally-finite.

Proof. Let R n denote the restricted Burnside conjecture.R has been 4 solved by Sanov (as mentioned above), R has been solved by G.l!igman 5 and A. I. Kostrikin. Later Kostrikin proved R for all primes p. This p

J

implies the truth of R , R J R J R and R (c f , M.Hall [9 ,p. 6 10 12 20 15 331). V.D.Mazurov [18J proved, that R holds. Our claim is a direct 30 consequence of these results, Q.E.D.

ULRICH FELGNER

304

We note that also all locally-finite, locally-solvable groups (i. e. groups in which all finitely generated subgroups

are

finite sol-

vable groups) of exponent 60 are uniformly-locally-finite (cf.M.Hall

[9 J,p.331). However in the case of solvable groups the following holds: Every so l.oab l.e group G of fini te exponen t is uniformZy...:z.oaa LZy-fini i:e , This follows readily from lemma 1 by a simple induction on the derived length of G. We close this section with some model theoretic remarks. lemma will give a useful algebraic characterisation of cal groups.

LENMA 3. Let

G

be any group of cardinaZity

::$

are equivaZent: (i)

is

G

ccc:

~o

For every group

H

of aardina Zi ty es

some u l t rafi l t e» D on ww, then (iii) {l}

G~ H ;

No . Then

N:o :

if

o

n

n+1

k E wand every finite e ub e e t: X

an automorphism

of cardinaZity

of G e uch that

1f

G=

G-C/D

If

U A nEW n tc ,

next

the foLZowing

~ H'C/D

for

An (nE w),

There is an increasing ahain of finite subgroups

= A ~ .•• S A S A . <;;. ••• , such that

The

N:.o - categori-

and for every there

X c. G,

is

(X) S;;; A

k

IXo - categoricity using Shelah's isomorphism theorem (cf. Comfort-Negrepontis [6],

Proof. (ii) is just a reformulation of the concept of pp.333-336). Whence

(i)~

(ii). If G satisfies (iii) then G has only

finitely many k-automorphism types. Hence G is necessarily ~ocate~ rical. It remains to prove that (i) implies (iii). Assume that G is

No - categorical and put A 0 = {I}. Assume for induction that A n- 1 is constructed. Let Cl 1 " " ,Cl m be the set of al~ n-au~omorphism-types and select from each of the Cl. one n-tuple: (x J , ••• x J ) E Cl • • Now let A* 1 n j n J be the subgroup of G generated by A _ and all the xi for 1::: i -< n n 1 and l~j Sm. Since G is locally-finite A* is a finite subgroup of n G. Let G = {g" ;" E.w } bean enumeration of G, and let An be the subgroup of G generated by A* V{g l}' It is easily seen that the n ngroups An do have the properties mentioned in (iii). Thus (i) implies (iii), Q.E.D. REMARK. The fact that

No - categorical

groups

are

uniformly - locally-

finite is an immediate consequence of lemma 3. In fact let in lemma 3 (iii). Card (X)

=

k

then

If X If

is a finite subset of G =

(X) So Ak ' whence

Card(X»

U nEW

A

n -< Card(A

A

n such

k).

be as that

STABILITY AND Ko-CATEGORICITY

305

DIGRESSION. A well-known theorem of M.I.Kargapolov and P.Ha1l-C.R. Kul atilaka [12] states that every infinite locally-finite group contains an infinite Abelian subgroup. The proof uses the theorem of Feit and

Thompson ("all finite groups of odd order are solvable"). It is, hence, worthwhile to mention, that the weaker statement: "Every infinite uniformly-locally - finite group contains an infinite

can be proved quite easily without using the

Abelian subgroup"

Feit-Thompson theorem.

In fact proceed as in Hall and Kulatilaka's paper [12]; then the infinity of ~(G) is already the final contradiction. An application of the Kargapolov-Hall-Kulatilaka the following

theorem yields

LEMMA 4. Let

Th(GJ

G be an infinite group and assume that

axiomatized by that

un1:versal

sentences

can

together with sentences

be

saying

G has infini te t» many e l emen t e , Then the fo l l ou i n q are equiva-

lent: (iJ

G

(iiJ

G

is ~ - categorical; o is an elementary-abelian p-group.

Proof. (ii) :=>(i) follows from Rosenstein's paper [20] . We shall prove that (i) implies (ii). It follows from (i) that G is locally-finite. Hence G contains an infinite abelian subgroup A. Since G has a fin i te exponent, A is the direct sum of cyclic

groups

of prime power

orders (cf. W.R.Scott [25] ,p.92). and only finitely many primes are involved. Whence A contains an i n f i.n i t e p-group B for some prime p , Then H B[pJ {x € B; x P 1 } is an infinite elementary-abelian p-

=

=

=

group. Since universal sentences are preserved under submodels and H is infinite, it follows that H 1= Th (G J, whence H == G • From this we conclude that G is elementary-abelian, Q.E.D.

§ 2.

DEFI NABLE

SUBGROUPS

A subset X of the group G is called a definable subset of G if G 1=

X = >{ gE.G;

age

~

par-ame t r-i oal.l.u definable if

formula

x={geG; GI=

(Xo'x 1, .. ,xn) E...r. and some

In general a group

G

for

some

a 1,· .a n EG.

has only a few definable subgroups. However,

ULRICH FELGNER

306 in the case of

~o-categorical

groups or stable groups the situation

is much more favourable. In fact we shall prove that the most important subgroups of an

~o

- categorical group are all first-order defi-

nable. We commence with the following observation:

THEOREM 1.Let

G be a aountably infinite

~-aategoriaal

group and H

be a subgroup of G. Then the fo l: lowing are equiva len t: (i)

H

is a aharaateristia subgroup of

(ii)

H

is a first-order definable subgroup of

Proof. Assume first that H a formula

G; G.

is first-order definable. Hence there is

x such that = {g E.G; G l=

(x) of ~ with precisely one free variable

H

peg)

=

'l'(x)

GI= 'l'[g)

a(g)

=

1T(g)

1T EAut(G) } •

Since Gis countably infinite

and

'l'(x)E.;'(..1},

p(g)=p(h)~a(g)=a(h)

holds

for

all g, h E G (to. see this use the proof of the Engeler-Ryll-NardzewskiSvenonius theorem given in Shoenfield [21] ,pp.91-92,proof of (c)=9(a»). G has only finitely many 1-types, whence {p(g); g ~G}

{peg) ; g E G} = {P 1 Svenonius theorem all the types p.

is finite. Put

, ... , Pm}. Again by the Engeler-Ryll-Nardzewski -

are formulas and

G 1=

3x

(

r.(x)E£1 such that J

r/ x)) •

are principal types. Hence there

GI=Vx(r.(x)~'¥(x))forall'l'EP" J J

Since H is a characteristic subgroup, H is the union of some of the 1-automorphism-types a(g) • But peg) p(h)~a(g) «( h ) , Hence

=

there exists follows that

T£;{l,2, •. ,m} such that H={gEG;

where

W .t€ T

jE T • Thus

r.(x) J

H

G

FW

JET

=

gEH~ 3jeT: p(g)=Pj' It

r.[g] } , J

denotes the disjunktion of the formulas

is a definable subset of G, Q.E.D.

r .(x) J

for

STABILITY AND

307

~o-CATEGORICITY

In connection with theorem 1 we should mention that an No-categorical group G has only finitely many characteristic subgroups. This follows readily from the finiteness of the number of l-automorphismtypes.(the countability of G is assumed here). One disadvantage of theorem 1 is that it does not provide us with a defining formula for a given characteristic subgroup. However, for many characteristic subgroups of ~o - categorical groups G of arbitrary cardinality we are able to give explicitely the defining formula. We shall do this in the sequel. We also

discuss parametrically

definable subgroups.

G be an No - c a t e qo r ic a L group and let T be a parametrically definable subset of G. Then (T) and (T G) are parametrically

LEMMA 5. Let

definable using the same parameters.

Proof. By the assumption T= {gEG; GI= 'I>[g.a .... a formula

1

'I>

and some elements "i ... ,an E.G. Put

fine by induction follows that

T

-r = {g·h ; gET n and hET n c;;;: Tn ~ T n + 1 ~ •••• Put

n 1 T~ To~ •••

U

D =

nEw

D = (T)

} .

n]}

for some ~

= TvU} Since

and de1E

To

it

Tn

Since G has a finite exponent (hence g -1 = gk that

To

for some k) it follows

Next define a sequence of formulas

'l>n

as

follows

:

let '1>0 (X o ' x 1 , ... X n ) be X = 1 v 1>(x o'x 1, ... x n) • If 1>n is defined, o then let 1>n+1(x o.x 1 •••• x n) be the following formula: 3y 3z [x

o

= y-z

1>n(y.x .... x 1 n)

I\,

I\,

1>n(z,x

1,

... X n ) ]

T = {ge G; G/= 1> (g.a •••• a ) } for all MEw. Thus m m n 1 implies GF1> (b.a , ••• a ),,-,1> (b.a ••• ,a ).But G has 1 1 m+ 1 m m+1 n m n only finitely many n+l-types. Hence T£. = Tt + 1 =... ,whence D = T9,.. Thus we have shown that (T) is parametrically definable by 1>i using

It follows that

beT

-T

a 1 ... ,a . n. G G G-1 Conc e rn i ng (T )put To = T v{l}, where T ={g tg; tET and gEG}, G) and proceed similarly as above. Then it follows that

m

In the sequel let ~

'I'

v=1 v

the formulas

'1'. v

of the x , 1-

(for

denote the conjunction 'P1 " 'I'2 m

1:s i ~ m). As usual

IT

i=l

x.

1-

A . . . i\

'I'm of

denotes the product

ULRICH FELGNER

308

LEMMA 6. Let G be an No-categorical group. Then there is a function y:w_w (depending only on G) such that for all subsets T of G, which are definable using n parameters from G, Y (n) ~ v=l

y (n)

{ Tf

(T)

v =1

Xv

(

X

v

=1

v

Xv

E. T ) } •

Proof. By the assumption T= {gEG; GF1>[g,a 1, .. ,a n]} for some Lformula 1> and some elements a1, .. ,anEG.Define T m and 1>m as in the proof of lemma 5. It was shown there that (T) z: T JI, for some JI" where JI, depends on a1, .. ,a n and 1>, thus JI,= JI,(a1, .. ,an,If». By the EngelerRyll-Nardzweski - Svenonius theorem G has only finitely many n-types.

Thus if ~(n) denotes the maximum of all numbers JI,(bl, •• ,bn'~)' where b1, .• ,b run through G and 1> runs through all ~-formulas containing n precisely n+l free variables, then ~(n) is finite. Hence T~( ) =

zy

COROLLARY 1.Le t

G be an N

-r

.J:.,- for* ** 1> (vo,··,vn),1f> (vo""'v

ea t'e q o r-i ca l: group. Then for each

o

mula If>(vo'v1, •• ,vn) there are ~-formulas n) such that for all "i : .. ,a EG : {xEG; GFIf>*(x,a1, •• ,a )} is the subn n G) = group of G generated by T={x€G; G!=1>(x,a .. ,a)} and

Proof. Using the notation introduced in the proof of lemma 5 the follows readily from lemma 6. In fact let 1>*(v ,v1, .. ,v) o n -formula: yen) yen)

following

at.

3Xl, .. ,XYCn,[Vo=D

Xv"

~1

(.1\,=1V1>(x V '

be

cl~m

the

V 1''''Vn))]·

1v Further, let ~(v -":> .. , v ) De 3w[lf>(ww, ":- •• ,v)1 ** 0 A* n o n If> (vo,v 1" .,vn):~ If> • Q.E.D.

and

put

COROLLARY 2. Let G be an ~-eategorical group. Then the commutatorsubgroup G' is definable (without parameters). Moreover all terms of the derived series G", G'" , ... ,G(r), ... and all terms of the lower central series are definable subgroups.

Proof. G' is generated by the set {[g,h]; g,heG}, where [g,h]is the commutator of g and h, thus [g,h] = g-lh-1gh. Similarly G" is generated by {[[x,y],[u,v]]; x,u,v,y€G},

etc. Thus G', G", .... are

ge-

STABILITY AND

309

~o-CATEGORICITY

nerated by first-order definable sets. Thus G' , G", . .• are also definable by the first corollary. For the same reason all terms of the lower central series (cf. M.Hall [91 ,p.150) are definable. Q.E.D. The Engeler-Ryll-Nardzewski-Svenonius theorem implies,that nearly all important subgroups of N'o - c a t e gor i c a I groups are definable subgroups. As in the case of lemma 6 this applies al s 0 to uncountable groups! As an example we mention, that if G is No-categorical, then the norm of G,

N(G)

= {a E G;

aH

=Ha

for all subgroups H of G}, is a

definable characteristic subgroup. In fact, since G has a finite exponent, N(G)={aEG;VgEG3kEZ: g-l a g=a k} has also a first-order

definition (cf.Schenkman, Illinois J.Math.4(1960)150-152). Following R.Baer an element x of a group G is called an FC-element if x G is a finite set. FC1(G)={XE.G; x is an FC-element Of G} is a characteristic subgroup and is called the FC-center of G • If G is No-categorical, then FC1 ( G) is a definable, locally normal subgroup of G such that the derived subgroup of FC is finite (Le. FC 1(G) 1(G) is a BFC-group (cf. D.J.S.Robinson [1~] p.126). An No - categorical group is not necessarily an FC-group. Consider the following group 3 2 G=Grp{ai,x; iEw" ai=x =1/\

x -1 aiaj=ajai"ai=ai i.

(where aX = x -1 ax; thus G is generated by {a.; iE W }v{x} with defining 3 2 x -1 1• ] relations a.a.=a.a., a.=x =1 and a.=a. ). Rosensteln [20 has shown 1- J J 1111that Gis N'o-categorical. Put H=

G be an

No

categorical group. Then the eoo l:e of G is a

definable charactersitic subgroup.

Proof. By the first corollary to lemma 6 there is a formula i\(v ,~) such that for every aEG: (a G) = {XEG; GFA[x,a]}. Thus (a GO )is a G) minimal normal subgroup iff

<

S={aEG;G!=VvlA(v,a)" v;il ~ \!y[A(y,v)_A(y,a)]]}

Put Soc (G) =

ULRICH FELGNER

310 § 3 • CHAIN

CONDITIONS

It is easily seen that an infinite uniformly-locally finite group is neither artinian nor noetherian (cf.[t9] p.38 for terminology) .In particular, an infinite No -ca tegorical group G does no t satisfy the minimality condition on abelian subgroups (since 1. G has a finite exponent, and 2. G has an infinite abelian subgroup). Thus it seems to be more natural to look at minimality conditions on dclinable subgroups. However an X - categorical group (of arbitrary cardinality) o has only finitely many definable subgroups (definable without parameters!). This follows readily from the finiteness of the number of 1-types. Chain conditions for parametrically definable subgroups are implied by No-categoricity only if the number of parameters is bounded. However, in this context another concept from model theory, namely that of A-stability,proves to be useful. M.Morley proved that a first-order theory T (in a countable language) which is categorical in some uncountable power is w-stable. w-stable theories are stable in all infinite powers A(c£. Chang Kei s Ler [5] p.404-405). A theory is stable if it A-stable for some infinite power A. e

THEOREM 2. An w-stable group satisfies the minimal condition for parametrically definable subgroups.

Proof. Assume, if possible, that there is an w-stable group G which does not satisfy the minimal condition for parametrically definable subgroups. Then there is a strictly decreasing sequence of subgroups H ~ H1 =.:l ... =.:lH :::::JH +1::>'" (new) such that for each nEw there .0 n n (n) (n) (n) i s a formula ¢n(Vo'Vl, .. ,vk(n~ and some elements ": , a 2 , .• , ak(n) in G such that: Hn

= {X€G;

rn ) a!=¢n(x, ": , ..

(n)

,ak(n) )}.

Since H + 1 is properly contained in Hn, each coset of Hn contains at n least two different cosets of Hn + 1. Let (x o ,xl'" ,x n_ 1) denote the ordered n-tuple of the objects x Using the axiom of depeno""xn_ 1. dent choices we see, hence, that for each sequence 0= (00'01 , ..,om_1) of O's and l's (i.e. 0i€{o,n for all O~i

HCb

(oo, .. ,om_2,om_1) m

and:

H

(oo, .. ,om_2) m-l

'

STABILITY AND b

where

xH

then put

Hf"'\b (ao, •• ,am_Z'O) m (a

m

= {x·h;

hEHmL

o,

.. ,a

m_ Z,l)

a

if

bftmHm ~ btrnH n

-1

"~n(bgtn'v,

Clearly xEG realizes Pf that P

m

":

(n)

n

m~

•

(n)

, .. ,ak(n))

t

f

iff XEn{bfl'mHm; mEW ,m~l}. I t

={a(n); nEw"l:::j~k(n)} u{b

follows

can be extended

f

(G, Y\EA

in the expansion

,.

w ; nEw" gE Z"f n=Fgl n}

is consistent with the theory of G. Thus P

to a (maximal) l-type ~

zt

Thus, by the choice of the e-

Now define for each f E. Wz -1 (n) (n) P f = {~n(bftn·v,al , .. ,ak(n»); n e c J V ~,,{

H=¢,

If f: w-{O,1} is any function and n

f~m= (f(O),f(l)' .• ,f(m-l)).

lements b

311

~o-CATEGORICITY

,where

aE.UnZL nElN Since A is a countable subset of G, and G is w-stable, G has, by deA

J

a

;

fini t i.ori , only countably many l-types over A. But ZNo , a contradiction. Q.E.D.

cardinality

{Pf

fEwZ}has

If G is an w-stable group and X<;;G, then there is a finite subset Y~

X such that

CG(X) = CG(Y). Thus centralizers of arbitrary subsets

are always parametrically definable.This is an immediate consequence of theorem 1 and the fact that

c ) = C (X v G(x1)ncG(x 2 G 1

X

2).

A strengthening of this fact is due to J.Baldwin (unpublished) .He showed that, if G is a stable group and nite subset

Y~X

X~

CG(X) = CG(Y).

such that

G, then there

is a fi-

In fact J.Baldwin shows

that [u,v) j 1 has the order property.This has three interesting consequences: (!) Let

G

be a stable group. Then the lattice

1c (G)

fc(G) = {CG(X); x£ G},

tralizers of subsets of G ,

of all cen-

satisfies

the

descending chain condition and the asoending chain condition.

~a (G) also satisfies the ascending chain condition, since the mapping X~ CG(X) is a bijection of ~c(G) which inverts the partial order.

(1; t) Maximal abe lian subgroups of stab l.e groups are parame trica lly definable.

In fac t, if A G, then (~ti)

A = CG(E)

is a maximal abel ian subgroup of the s table group for some finite subset

No categorical, stable group then there G such that CG(E) is an infinite maxi-

If G is an infinite

exists a finite subset mal abelian subgroup.

E~

E of A .

ULRICH FELGNER

312

(tt ~) is a direct consequence of (t t) and the theorem of Kar gapo lov-Hall-Kulatilaka (see § 1 ). LEMMA 8. Let G be an

~-eategorical,

stable group. If

N:j { l} is a

normal subgroup of G then there exists a minimal normal subgroup of

M

such that {1};i M S N .

G

Proof. Pick ": EN 51 G such that xl:j 1 . I t follows that

x

<

>,

•••

>-;j}

.... ~

<:J

N S} G

By the first corollary to lemma 6 there exists a formula such that

1}.

n «; m

~ ~

•

AC w :

v

o'

": )

V v [ -A (v ,xm ) => A (v, x n )]

/I

3 w [ A (w,xn )

Thus, by a result of S.Shelah [26J , theorem 2.13, stable, a contradiction. Q.E.D.

G

1\ . ,

A ( w ,xm )]

would not

be

Following R.Remak we define the socle of a group G to be the product of all the minimal normal subgroups of G, with the understanding that should G prove to have no minimal normal subgroups the socle of G is {l}. Let Soc(G) denote the socle of G. The upper socle series (or ascending Loewy series) of a group G is the ascending characteristic series {Sa} defined as follows:

If the upper socle series {Sa} reaches G in a finite number of steps (i.e. G= Sn for some nE e ) then G is said to have finite Loewy length. If G satisfies the minimal condition for normal subgroups,then G has finite Loewy length. We shall prove now that the same holds for ~ o categorical stable groups. This again supports the thesis that ~-ca o tegoricity and stability are strong finiteness conditions.

THEOREM 3. No-categorical stable groups have finite Loewy length.

STABILITY AND Proof. Assume that G I

313

~O-CATEGORICITY

I

l. By lemma 7 Soc (G) is a definable subgroup of G. Thus the stability of G implies the stability of Gj = G/Soc(G) , and the No-categoricity of G

{l}.

I t follows from lemma 8 that

implies the No-categoricity of

If Gj I t n

"s :

,

so c t q )

{l

then again by lemma

8 SodG 1) 1 O}, and hence {l}=So$ 81 = SodG)~ S2=Usoc(G/81J. I t follows by induction that Sn+1 I {l}, 8 ~Sn+1 provided that: n G/S n :j {1} (note that all terms of the upper socle series are def'inable) By the Engeler-Ryll-Nardzewski-Svenonius theorem G has 9nly finitely

many 1-types. If a E: Sn - Sn-1 and bE 8 n+1 - Sn ' then the types of a and b are different, since the groups 8 are definable. It follows n that G = Sk for some k c: w , Q.E.D. § 4 • NILPOTENCY

A group G is called loeally nilpotent if every finitely generated subgroup is nilpotent. If G is locally nilpotent and G:=: H , then in general H need not be locally nilpotent. However 'local-nilpotency' is a first-order property for No-categorical groups. More generally the following holds: LEMMA 9. Let G be a uniformly-locally-finite group and let be elementarily equivalent (G ( i ) If

(iiJ If

= H).

G is l.oc a lly-ni l.p o t en t-, G

Hand G

Then the following holds:

then

is locally-soluble, then

H is l o ca lly-ni l.p o t en t , H

is locally-soluble.

Proof. As mentioned. earlier (see § 1) the facts that

G is uniformly

locally finite and G =. H imply that H is uniformly locally finite • Ad (i): Notice first that a finite group A is nilpotent iff any two elements of coprime order commute (cf. Huppert: Endliche Gruppen, I, p. 260). It follows that the uniformly locally-finite group G is locally nilpotent if and only if any two elements of coprime orders are permutable. Let 0 (x) denote the order of x. Since G has a finite exponent there is a firs t-order formula Hence G

[(o(aJ,o(b))

=1

E (v 0' v ) 1

such that

fo r

any

=> ab =ba] is equivalent to cl= E(a;,b). is locally nilpotent iff cl= 'I x Vy : E (e, y). But also H is

elements a,b € G :

uniformly locally finite. Hence H is locally nilpotent iff HI= E(x,y). Thus the claim follows from G=H. Ad (ii): J.G.Thompson

l28]

\:Ix'ly:

has shown that a finite group is soluble

iff 'r/x'VyVz Lif x11, yl1

and

z l l , and the x,y,z have pairwise

ULRICH FELGNER

314 coprime orders, then

xy z t 1

J . This

condition is a firs t-order pro-

perty for uniformly locally-finite groups.

finitely gene-

Henc~ifall

rated subgroups of the uniformly locally-finite group G are solvable, and if G =H, then all finitely generated subgroups of H are solvable, Q.E.D. THEOREM 4. Let

G be an

No

-s

c a t e qo r-i oa L stable group. Then:

t i ) If G is locally-nilpotent, then G is nilpotent; (ii) If G is locally-soluble, then

G is soluble;

(iii) If G does not con tain e Lemen ts of order 2, then G is so l.ub leo

Proof. Ad (i): By theorem 3 G has finite Loewy length, Le. the upper socle series {Sa} reaches G in a finite number of steps.Observe that minimal normal subgroups of locally-nilpotent groups in the center (cL P.Hall

[10]

are

contained

,p.13, theorem 2.9). Thus

S1 = Soc(G) ~

Z (G)

Since S1 is a definable normal' subgroup GIS

1

is again an ~o - catego-

rical stable, locally-nilpotent group. Thus again Soc(GIS

1)

s;;

Z (GIS

1).

By induction it follows that the upper socle series is a central series. Since Sn = G for some nEw, it follows that G is nilpotent. Ad (ii): Proceed as above and observe that

in a

locally

theorem 2.9). Hence the factor groups of the upper abelian, whence G is soluble (cL M.Hall

[9]

soluble

[10] p , 1 3,

group all minimal normal subgroups are abelian (c f , P. Hall

socle series are

p.140).

Ad (iii): If G does not contain elements of order 2 then all finitely genera ted subgroups of G have odd order. Thus, by the Feit-Thompson theorem, G is locally soluble, and the claim follows from (ii),Q.E.D. REMARK. I t is not known whether locally-nilpotent No-categorical groups are necessarily nilpotent. However the following holds: (l)

Le t

G

be an No c a t eqo x-i ca l ,

l.o ca lly-ni l.p o t e n t group. If G has

only finitely many conjugacy classes, then G is a finite nilpotent group.

This follows from Kegel-Wehrfritz the fact that

[1't] ,p.44, theorem 1.H.4,

and

G clearly satisfies min-no

Definition (H.Wielandt): A subgroup H of G is subnormal in G (in symbols: H

H~~

~ A

G , then

2

-5J ...

~Ak

= G

1,

.. ,A

k

of

d(H;G) denotes the defect of H in G . Hence

G

STABILITY AND d(H;G)

z:

the least k such that

Following R.Baer

~

315

~o-CATEGORICITY

1 such that there are subgroups B

1,

••. ,B

H z: B ~ B ~ ••• ~ B/"S!B + ~ •• • ~Bk z: G . j 1 1 2

[1 ]

k

of

G

a group G is called a Nil-group if cyclic sub-

groups are always subnormal in G (i.e.

(g>

for all g E G).

G is

called upper-nilpotent if all nontrivial homomorphic images of G have nontrivial centers. G satisfies the Normalizer-condition if HI NG(H) for all proper subgroups

of G. Following Gruenberg a group

H

G is

an Engel-group if for all x,YEG the sequence Uo =x, .. ,un+1=[un,Y] reaches 1 in finitely many steps (i. e. Uk z: 1 for some k E w ) • It is well known that for finite groups

coincide.

No -categoricity

and stability

G

are

all

these conditions

finiteness

conditions.

Hence we shall show that for No-categorical, stable groups all variants of nilpotency coincide.

COROLLARY. If G is an N-categorical o are equivalent: (i) G is nilpotent; (ii) G (iii) G (iv) G (v) G (vi) G

stab l.e group, then the following

is locally-nilpotent; is a Nil-group; satisfies the Normalizer-condition; is upper-nilpotent; is an Engel-group.

Proof. (i) ... (ii) was proved in theorem 4. Subgroups of Nilgroups are Nilgroups (cf.Baer [1 ] p.406) and finite Nilgroups are nilpotent. This proves (iii) => (ii). But (i) The implication

~

(iii) is obvious; whence (i)

(iv) =>(ii) is due to Plotkin and

~

(i)=>(i~

(iii) •

is obvi-

ous. Claim: (v) =;> (i). Consider the upper central series {Zn}

where

Zo={l}, Zl= Z(G), Zn+l z: {xt:G;'Vye G: [x,yJEZ n}. If G is upper nilpotent, then Z (G/Z n) z: Zn+l/Zn I {1} provided G I Zn • Since G has only finitely many l-types, the upper central series becomes constant

at some finite stage. Hence G= Zk for some kEw, .i e , G is nilpotent. v

The converse (i)

~

(v) is obvious, whence (i)

~

(v) , (ii)

~

(vi) is

obvious. On the other hand Zassenhaus and Zorn have shown that finite Engel-groups are nilpotent, whence (vi)=;> (ii) since G is locally finite (d. Baer [1 ]P.407). Q.E.D. LEMMA 10. Let

G be an

finable normal subgroup

Ho

categorical stable group. Then G has a de-

N;:iil G euoh that

G/N

is nilpotent and N is

ULRICH FELGNER

316

the least normal subgroup of G such that

Proof. If P is a prime, then let by the set {g

E

Gp

be

GIN

the

is locally-nilpotent.

subgroup of G genera ted

o t q ) is a pOlJeX' of p} .Since G has finite exponent,

Gj

lemma 5 implies, that G is a definable normal subgroup of G . -1 -1 p [ A. B] denote the subgroup generated by {a b ab j a € A 1\ bE: B} put N

n

= p* q

[G • G

p

q

J = (\ p

where, for a set" of primes, G", all

a: E G such that no prime

Let and

G ,

p

is the subgroup of G generated by

q e « divides o I x ) , Thus N is the locally

nilpotent residual of G. By lemma 5 N is definable, and hence GIN is nilpotent by theorem 4(i), Q.E.D. In the next section (§ 5) we shall prove

that analoguously every

No-categorical stable group G has a unique maximal normal subgroup R such that R is nilpotent. Moreover every locally-nilpotent

normal

subgroup of G is contained in R. R is the Baer-Nil-radical of G and we prove that R is definable and coincides with the Fitting-subgroup and the Hirsch-Plotkin-radical of G • In general, an infinite group need not have a Fitting-subgroup(cf. W.R.Scott [ZS]p.166).

The

Fitting-

subgroup exists if the group satisfies the maximal condition for normal subgroups. Our emphasis is that also ~-categorical stable groups do have a Fitting-subgroup.

§ 5.

THE FITTING SUBGROUP

We introduce the following notation. If A and T are subsets of the T A = {a -1 ta j a € A 1\ t E T} J

group G , then

1t-1atjaEAI\ [A.TJ = the subgroup generated by{atET}. Obviously TA~<'TvLA.TJ> since a- 1ta=t(t-1a-1 t a)=t.[a.t]-1. Also [AJT]STA.T£T A• This proves that

LEMMA 11. Let G be an k E

III

~o

categoX'ical group. There exists an integer

(depending only on G) such that foX' all i f

d(g)j G )

g EG : ~

k •

N + =

Proof. For TeG define No(T)=G.

describing N.(T) is as follows: put

317

STABILITY AND Ko-CATEGORICITY is the least normal subgroup of N

which contains

G

More

T.

generally

is the least normal subgroup of Ni(T) containing T. This im-

i+ 1(T)

plies that d(T);G) ==j+l ~

(T)

z:

N.(T) J

provided (T)

5 all groups Ni({g}) ,for g € G , are parametrically definable. 1

Now let

be a defining formula of Ni({g}), i.e.

i(x,y)

Ni({g})

=={h€G;

GF1

iLh,g]}.

Notice that,by the first corollary to lemma 6, the defining formulas 1

of Ni({g}) do not depend on the parameter g. Thus if (g)

i(X,y)

and (g) has defect j+l in G , then (g)

N/{g})

z:

i.e·GF'VX[('l'.(X,g)~Wx==gT) " 3 V ( 'l" _1 ( V, g ) " J

m

where

J

't":o

and (g) 1" N _ ({g}), j

1

T) ] ,

&'- V 1" g

T=O

is the exponent of G. Since G has only

finitely

many 1-

types, it follows that there necessarily exists a bound k on the defects of the cyclic subnormal subgroups (g)

, Q.E.D.

THEOREM 5. Le t G be an No «o a t e qor-i c a l: stab l e q roup , Then oh ar-ac t e i-i e t i c subgroup R such that: (i)

R

is definable

(ii) I f

A

Proof. Put

(without paY'ameters)

R== {gEG;

(g)

R.Baer [1]

then

R

Hence

G •

A~ G

(a) -<:1<:1

A£; R.

has shown that

R

is a

R is a defi-

is an No-categorical stable group. By

the corollary to theorem 4 R is nilpotent. Now let normal subgroup of

a

(see Baer [1] ,p.418, Satz 2).It is

an immediate consequence of the proof of lemma 11 that nable subgroup of G, whence

has

and nilpotent;

is a nOY'mal nilpotent eub q r o up of G,

characteristic Nil-subgroup of

G

for all a € A, whence

A

£:

R

be a nilpotent

A

Subgroups of nilpotent groups

are sub normal.

by the definition of

R, Q.E.D.

The subgroup R==R(G)=={g€G;

c

al:

of

R==R(G)

G

(or: the

Baer-r

r

ad i ca l: in D.Robinson (19) p.61).

is intermediate between the Fitting subgroup

In

general

(if it exists)

and the Hirsch-Plotkin radical of G.

COROLLARY. Let subgY'oup of

G be

an No-categorical stable group. Then the Fitting-

G exists and coincides with the Bae r-r N'i.Lr- r a d i e a l: and the

HiY'sch-Plotkin-Y'adical

of

G~ pY'ovided

G is countable.

This follows immediately from theorem 1 (the

Hirsch- Plotkin radical

ULRICH HLGNER

318

of G , call it HP(G), is definable), theorem 4 (HP(G) is nilpotent) and theorem 5 (HP(G) S R'" R(G)) , since HP(G) is always a locally-nilpotent normal subgroup of G such that R(G)~ HP(G) (cf. D. Robinson

[t9] p.S8, p.61), Q.E.D. Note that without the countability assumption of G in the corollary we do not know whether the Hirsch-Plotkin radical HP(G) is definable. However the lemmata 5 and 6 are still available. Hence in every N o categorical stable group the Fitting subgroup exists and coincides with the Baer-Nil-radical. It is possible that the Fitting subgroup equals {1}. Let Fi t (G) denote the Fi tting subgroup of G. In the case of finite soluble groups G it is well-known that CG(FiUG)) S Fit(G). I t is interesting to note that CG(Fit(G)) ~ Ei.t t G) also holds in the case of locally-soluble N - categorical stable groups ( to see this use o theorem 5 (Fit(G) is a definable subgroup), theorem 4 (G is soluble), and W.R.Scott [25], lemma 7.4.6). § 6.

No -

CATEGORICAL SIMPLE GROUPS

We shall say that a group H is involved in the group G if H is a homomorphic image of a subgroup of G (i. e. HE: HSG). A group which is involved in G is also called a section of G . A theorem of Kargapolov [1~J states that in an infinite simple locally finite group there are infinitely marty non-isomorphic finite simple groups inVOlved. P.Hall and G.Higman [11] conjectured that, for each integer n, there are only finitely many isomorphism classes of finite simple groups of exponent n . Among those finite simple groups which are known up to now there are in fact only finitely many isomorphism classes of groups of exponent n (for each nEW). A positive solution to the Hall-Higman conjecture would imply (using the Kargapolov-theorem)that a locally finite simple group of bounded exponent is finite. In particular such a result, if true, would imply that No-categorical simple groups are finite. This would considerably simplify investigations concerning No-categoricity in the theory of groups.In the sequel we are, hence, only able to make a modest contribution to the problem of ~o-catego rical simple groups. We commence with the following observation. LEMMA 12. Let

G be an

No -categorical

is also a simple group (thus for ~o-categorical groups).

simple group. If

'simplicity' is a

H :G, then H

first-order property

STABILITY AND

319

~ 0 -CATEGORICITY

Proof. If G is a simple group, then G z: (xG) for all x E: G , x 11. By the first corollary to lemma 6 there is a formula ~(x,y) such that for all xEG

G be an No-categorical simple group.If G contains at

most finitely many elements of order 3 then

G

is finite.

Proof. Put T= {gEG; o(g)=3},where o I q ) denotes the order of g. By the assumption T is finite. Obviously T is closed under conjugatio~ Hence (T) is a normal subgroup of G . If G is infinite, then T) t G because (T) is finite. But G is simple, whence T z: f?J. G is No-categorical, hence locally finite, and by a result of Kargapolov [13] G has infinitely many non-isomorphic finite simple groups involved in it. These finite simple groups involved in G are homomorphic images of finite subgroups of G . Since T is empty, by Lagrange's theorem all these finite simple groups are 3'-groups (i.e. they do not contain elements of order 3). A theorem of J.Thompson states that such groups are Suzuki groups. But G has a bounded exponent.For a fixed exponent there are however only finitely many non-isomorphic Suzuki groups~ a contradiction. Thus G cannot be infinite, Q.E.D.

<

An infinite locally-finite simple group contains infinitely many elements of order 2 (otherwise, by the Feit-Thompson theorem,such a group would be locally-soluble and hence cyclic of prime order - cf. P.Hall [10) p.14). Hence, i f G is an infinite N - categorical simple . 0 group then there is a prime p~5 such that G contains infinitely many elements of order 2, infinitely many elements of order 3 and infinitely many elements of order p (and perhaps elements of further orders) , P.X.Gallagher [8] proved that if G is a finite group and G' its commun, tator subgroup and I G' I ~ 4 then each element of G' may be written as a product

of n commutators. Here /G'I denotes the cardinality of G'. In particular if G is a finite simple non-abelian group such that lGI~4n then: G = G' and hence every element of G may be written as a product of n commutators (the number n depends only on G). We shall prove a simi-

ULRICH FELGNER

320

lar result for products of conjugates in No-categorical simple groups. In the proof of this result considerable help came from U.Dempwolff which I acknowledge gratefully. We commence with an

elementary num-

bertheoretic lemma. Let a

.. ,a (m?2) be positive integers, not all zero, and assume that the 1, m greatest common divisor of all the a .. ,a is 1 , i.e. (aZ, .. ,a =1. There 1, m m) is an integer A ~ 1 such that for every x ~ A there are integers yi ~ 0 such

(ll)

m

that

x=L, a.y. i

z:

1

tr t:

Proof of (ll): We may assume that a 21 for all numbertheoretic lemma

i

(cLLandau [15j

possibly negative, such that solute value of zi and put

'£

~

a.z.

""

i

1~

~

m , By a wellknown

z., z,

p.31) there are integers z:

1. Let

.12:.

[s

0

"

denote the ab-

A= al'(£ a·lz·I)· i~ 1

Since 1

z:

t:

t.

the claim can be established quite easily for A, A+1,

L aiz

i, ... , A+n, etc. Q.E.D.

THEOREM 6.

Let

G be an No-categorical simple group. There

positive integer

y

y

'rJa€GVb€G

exists a

(depending only on G) such that 3x 1, .. ,x yEG[a*1

=\>

=Tr

b

x~lax.] "

i=l

'l-

Pro o f . Let a€. G and bEG, a 11, be given. Since bEG clear that D(a,b)

;m~l

m

:b=Tf y-.1 ay. m i=l"" is non-empty. Claim: the greatest common divisor of all mED(a,l) is 1.

Let that

1;

z:

{mEw

A

3y1, .. 3y

be the greatest common divisor of all 1;

mE

D(a,l),

and suppose

>1 . Let 2(1;) be the finite cyclic group of order

Dempwolff's trick and define a homomorphism

=.IT

T:

G

1;

•

We

use

-2(I;}as follows

for gEG put T(g) =u iff g y~lay. such that u:m (mod I;),for ,,=1 t. i: some Y1''''YmEG.We have to show that T is well defined.Notice first k=l, that I; divides ora), since if a then 1=U- 1a1)(Z-l a 1) .. ·U- 1a1 )

1; I k , Now, if n-1 e z: y. ay. z: IT z. zz a . t: t. i= Z t. t: for some y.,z. EG, then 1 z: (Tfy-.1 a y.)·(ITz-. 1az.)-1 is the product of

(k factors), whence

k € D(a, 1), whence

m 1T i= 1

t:

t:

-1

""" t: s: ora), whence

m « n t k -1) conjugates of a (where k

-1

(zi az

-1

i)

r

-1

==(zi az

i)

STABILITY AND where r=k-1.

It follows that m+n(k-l)E D(a,1J, whence, by the de-

finition of ~ : ~ 'm+n(k-l), i.e. mEn (mod

321

~o-CATEGORICITY

nk;: n-m (mod ~). But ~Ik ,thus

O. This proves that '[ is well defined.

Clearly '[ is a homomorphism, thus

Z(~) ~ GIN,

where

is

N

the

kernel of '[ . This, however,contradicts the simplicity of the noncommutative group G. This proves that

~

=1. n tb i

For bE B let n i b ) be the least element of D(a,b), thus

s: 1.

It is an immediate consequence of the Engeler-Ryll-Nardzewski-Sver

nonius theorem that {n(b) ; bEG} is bounded (G

has only finitely

many 2-types!). We may define therefore:

n t G)

z:

the maximum of all numbers n t q ) for g E G •

Since, as we have seen earlier, the greatest common divisor m€

o t «,

By (Li)

of all

is 1, there are m , .. ,m E tit «, 1) such that 1 = (m , · · · ,m 1 t 1 t). there exists A ~ 1 such that for all x ~A there are integers

1)

r

s.~O such that

x= ,« . • Put f =n(G)+A. Since f

every element bEG can be written as a product of faconjugates of a. In order to conclude the proof let rent fa

y

be the product of all diffe-

(where a E G). Notice that, since G has

only

finitely many

1-types, there are only finitely many different numbers fa for aEG. Now, for arbitrary a EG, bEG, a 11, it follows that wOfa some w. and

Furthe~

b can be written as a product of fa

1 can be written as a product of fa

that

=

for

y

conjugates of a

conjugates of a. It follows

b z: b- L: ... · 1 (w-l copies of 1) can be written as a product of

y conjugates of a,

Q.E.D.

Note that in theorem 6 it is understood that G is non-abelian! Al though it is apparent now that

No-categorical simple groups ressemble

finite groups in many respects, it still seems to be a difficult task to prove that they are finite.

If tli.is is not the case, then

there

are still infinitely many finite simple groups to be discovered.

§ 7.

SOME

STRUCTURE

THEOREMS

The following lemma is more or less contained in Rosenstein

LEMMA 14. Let

G be a direct sum of finite groups

G is N-categorical then almost all K. are abelian. o

~

K.

~

,

G z:

(20) .

EB K ..

ieI ~

If

322

ULRICH FELGNER

Proof. G is clearly an FC-group. Hence the No-categoricity of

im-

G

plies that G is a BFC-group (Le. there exists a bound on the cardinalities of the conjugacy classes). By a theorem of B.H.Neumann (cf. D.Robinson [19],p.126) G' is finite. Hence at most finitely many of the direct summands K i LEMMA 15. Let

oN -categorical group and assume that G = EBK .. o . I 'IK. is a minimal normal subgroup of G such 'l-E.

G be an

If for every iE I, that

(for i € I) are non-abelian, Q.E.D.

'I-

is a non-abelian simple group, then

K i

Proof. Suppose, if possible, that I

I

is finite.

is infinite. We claim that

(n) If g=t1y2· ... ·yn = zlz2· ... ·zm 11 is such that all

and Yi = z1I(i) for some permutation

11

This is a consequence of the generalized Krull-Remak-Schmidt theorem

G

(cf. Huppert, Endliche Gruppen 1,1967, p.69). Since sum of simple, minimal normal subgroups K written as a product g =x

... ·x

i

is a direct

each g € G , g i l ,

such that

K.

can be

(for some i

1x2· n J 'Idepending on x.). As in the proof of lemma 7 we see that each K. J

'I-

is

parametrically definable. Thus, by (n), for each [lEG, gll,there is precisely one n such that g =x

•• ·x for some x , having the proper1· G n 'Ity that (x.) is a simple normal subgroup of G, and the correspondence 'I-

g~

n can be implemented within the first-order language

of group-

theory. But G has only finitely many 1-types. It follows that there is a bound k on those numbers n Q.E.D. THEOREM 7.

Le t

=n g

. Hence I

has at most k elements.

G be an No

There

finite chain of characteristic, definable subgroups Hi nEw)

(i~

exi s t s

a

n for some

such that {l}

= Ho~H1-:5!

... ~Hi-s1Hi+1~ ... -:5!H n

G,

and for all i

Proof. By theorem 3 G has finite Loewy length, i.e. the series So

= {1},

upper socle

Sl=Soc(GJ, ... reaches G in finitely many steps,

i e , G =Sn for some nE: w . Put H: = Si for i~ n • Each Si+1/Si is a direct sum of minimal normal subgroups D . . . Hence all groups D. . v

'I-,J

'I-,J

STABILITY AND

323

~o-CATEGORICITY

are characteristically-simple. By lemma 15 D . • is either abelian or "'>J a direct sum finitely many simple groups. Thus if we apply lemma 15 once more to SOliS. it follows that S. liS. = A. El1 B., where A. is ",+ '" ~+ ~ '" '" '& abelian and B is a direct sum of finitely many simple groups. Put i H0 z: {1} > H1 z: A 1 > H2 z: S 1 > H3:= U A2 > H4:= 2 > • • • then the claim of the theorem follows, Q.E.D.

Us

COROLLARY. Let G be an No-categorical stable group. If G contains at most finitely many elements of order 3 then there exists a finite

chain of no rma l , p ar ame t r i oa l l u definable e ub q r oup e Hi (iSn for some nEw»

11} :=Ho~Hl...sJ ... ~Hn:=G such that for> all

either> abelian

01'

i

is

a finite simple group.

Proof. Use lemma 13 and theorem 7, Q.E.D. REFERENCES

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