INFINITE GROUPS WITH SUBNORMALITY OR
DESCENDANCE CONDITIONS Ol‘i THEIR INFINITE
SUBGROUPS, AND SOME SPECIAL CERNIKOV p—GROUPS

Thesis for the Degree of PhD.
MICHIGAN STATE UNIVERSITY
VERIL LEROY PHILLIPS
1975

This is to certify that the
thesis entitled
INFINITE GROUPS HIT}! SUBNORMALITY 0R DESCENDANCE
CONDITIOBB ON THEIR INFINITE SUBGROUPS, AND SOME
SPECIAL éERNIKov p-GROUPS

presented by
VERIL LEROY PHILLIPS

has been accepted towards fulfillment
of the requirements for

LLdegree in ALA/471 (5‘

2.5 fig 43;

Major professor

Date W 73)-

0-7 639

 

ABSTRACT

INFINITE GROUPS WITH SUBNORMALITY OR DESCENDANCE CONDITIONS
ON THEIR INFINITE SUBGROUPS, AND SOME SPECIAL EERNIKOV p-GROUPS

By 0

Veril LPRoy Phillips

In [2], R. Phillips provides a characterization of groups in the
class LEI, n IJ‘YS - :18, of infinite, locally finite groups, with all
infinite subgroups serial, but with some finite subgroup not serial.
This class turned out to be equal to a similarly defined class by
Eernikov [1].

In this thesis two additional similarly defined classes are
studied: L?! I] Iflsnb - nsnb is the class of all infinite locally
nilpotent groups having all infinite subgroups subnormal and with a
bound on the defects of infinite subgroups, but with the finite sub-
groups failing to have that property. I." n Il'ld - 01d is the class
of infinite locally nilpotent groups having all infinite subgroups
descendant but some finite subgroups not descendant.

After reducing the study of groups in these classes to the study
of p-groups in these classes, necessary and sufficient conditions
for an infinite p-group P to be in the classes are developed. In
both instances these conditions include: P has a normal divisible
abelian subgroup D of finite rank, whose centralizer C has finite
index (not I) in P and furthermore for all x e P-C, conjugation
by x is an automorphism of D which does not normalize any infinite

proper subgroup of D. Such automorphisms are called S-I auto-

morphisms.

Veril LeRoy Phillips

By using the fact that the automorphisms of D are essentialLy
r X r matrices over the p-adic integers with determinant a unit,
it is further shown for the classes studied that the rank of D is
p - l, and the rational canonical form for S-I automorphisms is
computed.

Finally, using the study of 8-1 automorphisms, a structure
theorem for direct limits G of p-groups of maximal class is developed
which shows G is a semi-direct product of a divisible abelian p-group
D of rank p - l by a cyclic group of order p or of order p2
with an amalgamated subgroup trivial or of order p, respectively.

A relationship between these groups and the class LY! n stnb - nsnb
is established and examples thereby provided.

All known examples of groups in either Lu n Iflsnb - nsnb
or LI! [1 Ind - fld satisfy Min, hence are Eernikov groups. Indeed,
it is shown that a group G in the former class necessarily must
satisfy Min, but the corresponding question for the latter class

remains unsolved. If they also must satisfy Min, then the two

classes are shown in fact to be equal.

References

l. Eernikov, S. N., Groups with prescribed properties for systems
of infinite subgroups, Ukrainian Math. J. 19 (1967), 715-751.

2. Phillips, R. E., Infinite groups with normality conditions on
infinite subgroups, Rocky Mountain J. of Math., to appear.

INFINITE GROUPS WITH SUBNORMALITY OR DESCENDANCE CONDITIONS
ON THEIR INFINITE SUBGROUPS, AND SOME SPECIAL CERNIKOV p-GROUPS

By

Veril LeRoy Phillips

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1975

To Pam and RoiAnn

whose patience and understanding have been so supportive

Ho
Ho

ACKNOWLEDGMENTS

The author is naturally indebted to a number of professors,
but particularly to his advisor, Professor R. E. Phillips, for his
guidance and encouragement, and to Professor Lee Sonneborn for his
accessibility to discuss mathematics and to share his advice. In
addition I am grateful to Professor Robinson whose two volumes, often

cited in this dissertation, greatly assisted this neophyte.

iii

Chapter Io

Chapter II.

Chapter III.

Chapter IV.

Chapter V.

Chapter VI.

Bibliography

TABLE OF CONTENTS

Introduction and Notation. . . . .

Preliminary Results. . . . . . . .

Structure of L87 fl Iflsnb - nsnb

Structure of L8? fl Ik‘ld - X‘fd . .

Strongly Irreducible Automorphisms

Relationship to Direct Limits of p-Groups of

Maximal Class. . . . . . . . . .

iv

Page

22

35

M6

57

Chapter I

Introduction and Notation

In this paper we investigate two classes of infinite periodic
groups: The first has all infinite subgroups subnormal and with a
bound on the subnormal defects but with this condition failing for
finite subgroups. The second has all infinite subgroups descendant
but has some finite subgroup not descendant. We obtain a structure
theorem for the first class in Chapter III and a structure theorem for
the second class in Chapter IV. In Chapter V we develop some special
notation to discuss the kind of automorphisms of divisible abelian
p-groups which arise in Chapters III and IV, and use this notation
in Chapter VI to obtain a characterization of direct limits of p-groups
of maximal class and also to obtain more information about the first
class of groups. In this chapter we list our notation and in Chapter

II we state some preliminary results.

Notation 1.1: If F is any set, IFI will denote its cardi-

nality.

Definition 1.2: A class 2£_groups, X , is any collection

 

of groups such that whenever G e X and G is isomorphic to G

1
then also G1 6 X . G e X is also expressed by saying that G is

an X -group. Many authors require that a class of groups also contains

a trivial group (group of order 1).

Notation 1.5: If G and G1 are groups, G‘; Gl will denote

that G is isomorphic to G1.

Notation 1.h: The identity element of a group and a trivial

group will both be denoted by l.

NOtation 1.5: If X and Y are any sets (including classes
of groups) then X - Y denotes the set of all elements of X which

are not elements of Y.

Notation 1.6: If G and G1 are groups, their direct sum is

denoted by G + G If {Gill 6 I} is a collection of groups, their

1.
direct sum is denoted by §:[Gili e I}.
Definition 1.7: If G has a normal subgroup H, and H has

a proper supplement K in G (meaning that K is a proper subgroup

of G and G=HK), then G is a semi-direct productgf H b g

 

with amalgamated subgroup H n K. In case H n K is trivial, then

 

G is simply called the semi-direct product of H by K or a split

extension of H by K, and this fact is denoted by G = H]K.

Definition 1.7 provides the internal criteria for a group G
to be a semi-direct product with an amalgamated subgroup. In Chapter
VI we use Gorenstein's construction [2h; pp. 27-28] of the external
semi-direct product with an amalgamated subgroup. (Gorenstein's term

is "partial semi-direct product.")

Notation 1.8: " E_", " < ", and ” i." will denote, respectively,
"is a subgroup of," "is a proper subgroup of," and "is not a subgroup

of." These symbols will also have their usual meanings.

Notation 1.9: If X is any subset of a group G, then < X >
denotes the subgroup of G generated by X. < x, y, ... , z > will

be used instead of < (x, y, ... , z} >.

Notation 1.10: If G is a group, and H is a subgroup of G,

then

(i) lcznl denotes the index of n in G,

(ii) H Q G (H N G) means that H is (is not) a normal subgroup

of G,

(iii) H char G (H char G) means that H is (is not) a character-

istic subgroup of G,

(iv) NG(H) and CG(H) denote, respectively, the normalizer
in G of H and the centralizer in G of H; CG(x)

will be used in place of CG(< x >), and

(v) H is a Sylow p-subgroup (Sylow p'-subgroup) if H is

maximal in G with respect to all its elements having

order a power of p (prime to p).

Definition 1.11: In any group G and for any subgroup H,
there is a natural homomorphism f:NG(H) 4-Aut H (the automorphism

group of H) with kernel CG(H). (See, e.g., [M1, Thm. 5.2.5].)

 

Notation 1.12: If H is a subgroup of a group G, h e H, g e G,

and. a e Aut G, then

(i) hg = g-lhg:

(ii) hg is the image of h under a,

(iii) Hg is the conjugate of H by g,
(iv) H3 is the image of the subgroup H under a,

(v) HG = < hglh e H and g e G >, the normal closure of H

in G, and

<
(vi) H a>=<h6|heH and ae<a>>.

Our use of the term "series" corresponds to the definition of

"normal system" by Kuros [31, p. 171].

Definition 1.15: Let H be a subgroup of a group G. A series

 

between H. and .9 is a chain of subgroups .3 such that (i) if
XeS, then ngge, (ii)He,3,Ge3, (iii)3 iscomplete,
i.e., contains all unions and intersections of its members, and

(iv) if X has an immediate successor Y in the natural ordering of

.3, then X G'Y. ‘We write H ser G (H sér G) if there exists (there

does not exist) a series between H and G. A descending series

 

between H and G is a series .3 between H and G in which the
reverse of the natural ordering of .3 is a well ordering. H is
(is not) a descendant subgroup of G, written H desc G (H desc G)
if there exists (there does not exist) a descending series between
H and G. If H desc G, then a descending series .3 between H
and G will be denoted z = {Hula < o}, where 0 is an ordinal

number and where a <‘a < 0'==> H < H .
1 2 —ozl

Definition 1.1M: A descending chain of subgroups of a group G

 

 

is a chain (: of subgroups in which the reverse of the natural ordering

of C is a well ordering. Notice that a descending chain differs from

a descending series in that the former has no condition of normality

and is not necessarily complete.

 

Definition 1.15: A subgroup H of a group G is subnormal,
written H d'd G, if there exists a firite descending series

between H and G.

Notation 1.16: Let H oe a subgroup of a group G and a

an ordinal number. We define the subgroups HG’a inductively as

. G,d
G,O G,Ot+1=H(H ), and Ham: fl He,a

oz<X

follows: H = G, H for

a an ordinal and X a limit ordinal.
It is well known that H desc G if and only if there exists

G’a, and that in this case [HG’BlfB < a + 1}

an a such that H = H
is the fastest descending series between H and G. (See [55, Sec. 1.h]

or [57, Sec. 5].)

Definition 1.17: If H <l<lG, the subnormal index or defect
G,d

 

 

of H in G is the least nonnegative integer d such that H = H.
If H is a subgroup of G and s a nonnegative integer we write

H‘Glds G if H is subnormal with defect at most 3.

Definition 1.18: If x and y are elements of a group, the

 

commutator of, x_ with y_ is [x,y] = x-ly-lxy. If X and Y are

 

subsets of a group, the commutator subgroup of_ X_ with Y. is [X,Y] =

 

 

< [x,y]:x e X, y 6 Y >. If x1, x2, ... are elements of a group,
we define more general commutators inductively by [x1] = x1 and
[x1, ... , Xn+l] = [[x1, ... , xn], Xn+l]° If X1, X2, ... are subsets

of a group, we define more general commutator subgroups inductively

by [xl]=<x > and [X1, ,x

n+1

1: [[X1, , Xn], x ].

1 n+1

If G is a group, the derived group is G' = [G,G].

Definition 1.19: Let G be any group. The center of G is

 

denoted by Z(G). The upper central series of G is the series

 

{a3(G)l defined inductively by

G) = 1, z (G) = Z(G/Za) and z (G) = U Za(G) ,

+
a l A, a < A

O(

for (1 an ordinal and A a limit ordinal. The lower central series

 

of G is the series {7a(G)} defined inductively by

71m) = G, 7on1“) = Mamba], and me) = O‘QA 7am) .

for <2 an ordinal and I a limit ordinal.

Definition 1.20: Let G be a group. We will use the following

 

notation for classes of groups encountered frequently:

G e 3 if G is finite.

G e b‘( if G is nilpotent.

C)
m

31c if G is nilpotent of class no more than c.

G e Z if G has a central series.
G e ZA if G is hypercentral; i.e., if ;a(G) = G for some
ordinal on.

G e ZD if‘ 7a(G) = l for some ordinal a.

G e rrsnb if all subgroups are subnormal with bounded defects;

i.e., if there is a positive integer n such that

HG,n

G e :[d if all subgroups are descendant.

= H for all subgroups H of G.

G e #8 if all subgroups are serial.

G e Iflsn if G is an infinite group in which all infinite
subgroups are subnormal.

G e Idisnb if G is an infinite group in which all infinite
subgroups are subnormal with bounded defects.

G e 117d if G is an infinite group in which all infinite
subgroups are descendant.

GeIwS

if G is an infinite group in which all infinite
subgroups are serial.

G 6 L3 if G is locally finite.

G 6 L37 if G is locally nilpotent.

G 6 Min if G satisfies the minimal condition on subgroups.

G is a Cernikov group (extremal in Cernikov's works) if G

is a finite extension of an abelian group A 6 Min.

G is divisible (radicable in [58] and [59]) if for all positive

integers n and for all x e G, there is a y e G such

G is a group of type p” for p a prime if G is a noncyclic

abelian p-group all of whose proper subgroups are cyclic.

(See [21], p. 65.)

Notation 1.21: If G is a group and p a relevant prime,
then G(pn) denotes the subgroup of G generated by all elements

of order pn.

Definition 1.22: If G is an abelian group satisfying Min

 

then it decomposes into a direct sum of finitely many summands, each

of which is finite cyclic or of type p°° for some prime p. (See,

e.g., [22, Thms. 5.1 and 25.1].) The decomposition is not unique,

but the number of summands is an invariant, called the rank of ‘G.

 

Notation 1.25: Let G, G be groups, R a ring with unity,

l

and r a positive integer.

End G is the set of all endomorphisms of G.

Aut G is the set of all automorphisms of G.

Hom (G’Gl) is the set of all homomorphisms from G into G1.

GL(r,R) is the set of all r x r matrices over R and with
determinant a unit in R.

E is the ring of integers.

E? is the field of integers modulo p.

5 is the field of rational numbers.

R is a principal ideal domain (PID) if R is an integral domain

 

in which all ideals are of the form aR, for some a e R.
R is a unique factorization domain (UFD) if R is an integral
domain in which each nonzero element is either a unit or
can be written as a product of finitely many irreducible
elements of R, uniquely, up to the number of factors

and up to associates of the irreducible elements.

Definition 1.2%: An ordinal number N is cofinal with a limit

 

ordinal <1 if X is the limit of an increasing<1-sequence; i.e.,

if x = 1im m(§) for some sequence 9 of ordinals satisfying
F <‘a

.3

£1 < 52 <‘a implies ¢(§l) < ¢(§2) .

(See [29, p. 256].)

Chapter II

Preliminary Results

In this chapter we compile some general theorems, most of which
are well known, that we will use later. More specialized results will
be stated (with or without proofs) as they are needed. We begin with
some lemmas involving descending chains of subgroups, for which the
first requires the following result about ordinal numbers, which we

state without proof.

Proposition 2.1: [29; Thm. 2, p. 2h5.] Every limit ordinal

 

of the form A = lim o(g), where w is any sequence of ordinal
§<oz
numbers, is cofinal with some ordinal 7 E'a.

The following is a generalization of [15; Lemma 1].

Lemma 2.2: Let G be any group and [HGJG < o] a descending
chain of nontrivial subgroups with rl{Hdla < O] = l (and hence o
is a limit ordinal). Let F be a subgroup and u = mintB:[B[ 2 [Fl].
We suppose that for all limit ordinals X such that X E_u, o is
not cofinal with N. (This is satisfied automatically if F is finite.)

Then if' HdF is a subgroup for all a < 0, then flffldI10 < O} = F.

Proof: Let F = {fBIO 5.6 < u] be a well ordering of F with
f0 = 1. Let HO = Iltfldla < O] = 1. NOw for every B, O E_B < u

let e(s) be minimal such that

[f7l037gelnn {l} .

MB) 2

10

We claim that there is some ordinal g < G such that F.n Hg = 1.

Let g = lim cp(s). Note that g g o, and that F n Hg = 1.
B < H

Case 1. F is finite. For every B, O S'B < u, let 6(6) be the

minimal ordinal such that fB é H9( Note that 9(a) < o for all

B)’
5. Clearly

t=mnmmmgb<ul
and thus g < 0 since 0 is a limit ordinal.

Case 2. F is infinite. By 2.1, g is cofinal with some ordinal
A 3.“. If g = o, the hypotheses are contradicted, and so we have

g < o, as desired.

Next we claim that n (HaF) _<_ F. Let x e n (HaF).
g S'a < o g 51a < o

For all a, £50: < 0 write x = hafoc’ ha e Ha, fa e F. Then

- -l
= _ = . = < < O
hglha fgfix c HEIW F 1, hence h hg e Hg for all. a, g __a o

05
Ha: fl Ha=HO=1 and x=hf =f§eF,

Thus h e
§§_<_Ot<o oz<o gg

as desired. But it is now clear that

F g n (HaF) = fl (HaF) 3 F .
a < o g 51a < o

m 2.5: Let G be any group and H = (H7IV e P} any nonempty
collection of infinite subgroups with finite intersection. Then there
is a collection .L of infinite subgroups with finite intersection
with the properties: (i) for every L e at there is a subset A E I‘
such that L = (ltH7I7 e.A] and (ii) either .C has exactly two members

or .C. is an infinite descending chain.

11

Proof: For every subset A _<_ P, define LA = n {Hyh e A}.
Consider the set of all possible intersections of the members of ‘H :
2 = {LAM E P}. By hypothesis, LP is finite. Let A _<_ P have

minimal cardinality such that LA is finite.

Case 1. A is finite. Let A = {71, 7,), ... , 7n]. Then

LA = H710 1172/] (1 H7?1

is finite. Let m be minimal such that H7 fl “° 0 H7 is finite.
l m

Note that m 32. Take A1 = {71, 72, , 7 l and A = [7m].

m—l 2

By the choice of m, LA is infinite, and clearly L = H is
1 A2 7’m
infinite, but L131“ LA = LA is finite. We are finished by taking
2
L = {LA 3 L132}-

1
Case 2. A is infinite. Let G be the minimal ordinal of cardinality
[A]. Let A be well ordered with order type 0. Set I. = {LA'IA'
is a proper initial segment of A]. For every proper initial segment
A' of A we have [A'I < IA] by the minimality of G and hence

L

A' infinite by the choice of A. Furthermore, A' E A" _<_ A implies

LA" < LA' so that L is a descending chain of infinite subgroups

whose intersection LA = (LC is finite.

Lemma 2.h: If H and K are infinite normal subgroups of a

 

group G, HI) K = 1, and F is a finite subgroup of G, then

HF n KF is finite.

Proof: Suppose that HF n KF is infinite. Then there is an

f e F and an infinite collection {hi 6 Hli = l, 2, ...} such that

l2
hifeI‘EFr‘ICF, for all i=1, 2, .00

For all i = 1, 2, ... , let f. e F, k. e K such that h.f = k.f..
1 l 1 l 1

Since F is finite there is an infinite subcollection of the ki's

for which the corresponding fi's are all equal. Hence after renum-

i = 1, 2, ... and 1 ¥ 3 implies

berlng, we have hif = kifl’
hi £ hj' But then, ff;1 2 hglki e HK = H + K and the unique repre-

sentation of an element in H + K is violated. Hence HF n lCF‘ is

finite.

Next we collect some conditions for subnormality and descendance
in groups. The first is due to Robinson [55; Sec. l.h] or [57; Lemmas

5.11 and 5.12].

 

Proposition 2.5: If G is any group and H E.G, then H Q'er

if and only if [G, H, H, ... , H] E.H°
<----r ------ >

Sketch of proof: For the full proof, see the reference above.

It is easy to show that

HG’i = H[G, H, H, , H]
<----i ------ >
for each nonnegative integer i by use of induction and the usual
commutator formulas.
It is not generally true that descendant subgroups remain
descendant in homomorphie images, as the following example shows.

However, when the descendant subgroup contains the kernel, then it

remains descendant in the image as seen in 2.7.

15

Example 2.6: Let G be the dihedral group on an infinite cyclic

group. write G = < x >]< y >, where < x > is infinite cyclic and

y is an element of order 2 with xy = x-l. Let P be prime and
let
n
Xn = < xp > E’G for all n = 1, 2, ...

co 00

Notice that Xn <1G’ for all n, and [1 Xn = 1. By 2.2, [1 Xn < y >
n=1 1

:<y>. NOW

00
< >:-. < ><l°'°<l i <1 < <1": =
y 9 Xn y Xn+1 y> Xr1 y> <le<y> G

showing that < y > desc G. For all n = l, 2, ... let 9n:G-+ G/Xn
be the natural map. For n 2.1, < y > 9n = Xn < y > < G/Xn. We
claim that for odd p, Xn < y >/Xn is proper and self normalizing in

G/Xn° To see this, suppose otherwise. Then

< > < < .
Xn y NG(Xn y>)

Let z e NG(Xn < y >) - Xn < y >. Write z = meJ where m is an

integer and j = O or 1.

m
Case 1. j = 0. Then yX = x- my 6 Xn < y >. Hence x2m e Xn’

giving 2m = kpn, for some integer k. But pn is odd, so that

k = 2! for some 2 and hence m = Z . pn. But then 2 = x =
n
P)2

(x e Xn E’Xn < y >, contradicting the choice of z.

xm -2m 2m
Case 2. j = 1. Then y y = (x y)y = x y e Xn < y >. Thus

2
x m e Xn’ and we complete the argument exactly as in Case 1. Thus,

Xn < y >/Xn is indeed proper and self normalizing in G/Xn for p

1h

X < y >
odd. Now if j‘f— desc 3-?- then since G/Xn is finite,
n n
X < y >
—Lf'—— <1 <lr 32—, for some positive integer r, and hence
n n

< <1
Xn y > qu :
whence Xn < y > is distinct from its normalizer in G. Thus we are
assured that

<
Xn y>

G
X dasc f o

n n

Thus, although < y > desc G, we have < y > an de’sc Gen.

Proposition 2.7: Let G be a group and H, K, and L subgroups.

 

(i) If K<1G and HdescG with K_<_H, then

H G
H desc K .

(ii) If H desc K and K desc L, then H desc L.
(iii) If H desc K, then H n L desc K I] L.
(iv) If H desc G and H 5 K, then H desc K.
(v) If H desc G and K desc G, then H n K desc G.
(vi) The class Ind is closed under the taking of infinite

subgroups.

Proof: (1) Let 14: [Hula < o} be a descending series between

H and G. Then for all a < G we have K E H E Ha, so that also

H
%< 3%. Thus L = {Ha/Kla < o} is a descending chain containing
_ <1 < 0.
K an satlsiylng Ha+1/_K Ha/K for all on Let X be a limit

l5

ordinal, K _<_ G. Then there exists some L e H such that
L -.= (“Hello < x}. Thus L/K e .6 and it is clear that
L/K = ”(HO/K[a< X}. Thus L contains all intersections of its
members and since 3 is a descending chain, it contains all unions
of its members; i.e., L is complete. Thus at is a descending series
between H/K and G/K.

(ii) Let (H = {Hula < 0} be. a dwscc-L'i'lg series between H
and K and ‘H: {KB]B< X} a descending series between K and L.
Then WUH is clearly a descending series between H and L,
showing H desc L.

(iii) Let [Hala < o] be a descending series between H and
K. Then {Ha n Lla < o] is a descending series between H n L and
K n L.

(iv) Let HdescG and HEKEG. By (iii) we have
HflK=HdescGnK=K.

(v) Let H desc G and K desc G. By (iii) we have
H n K desc G nK = K. Thus by (ii) it follows that H n K desc G.

(vi) follows from (iv).

Proposition 2.8: (i) If G e Ind and K <lG, then G/K e Ind

 

if G/K is infinite. (ii) If G e Ind and H is an infinite normal

subgroup of G, then G/H 6 nd.

Proof: (1) Let H/K be an infinite subgroup of G/K. Then

H is an infinite subgroup of G and hence by hypothesis, H desc G.

By 2.7(1) we have [Eidesc %. Thus % e Ind.

16

(ii) Let L/H be a subgroup of G/H. Then L is an infinite

subgroup of G and by hypothesis we have L desc G. By 2.7(i),

L . G . G d
'H desc H) shOW1ng fi-e J1 .

Remark 2.9: We will use the ideas of 2.7 and 2.8 in later chapters

without reference.

Next we collect some elementary properties of Cernikov groups,
since they play a prominent role in our investigations. Recall a
Cernikov group is a finite extension of an abelian group with Min.

In particular all finite groups are Cernikov groups.

Proposition 2.10: (i) Every group G has a maximal normal

 

divisible abelian subgroup D (not necessarily nontrivial). (ii) If
G is an infinite Cernikov group, then G has a maximal normal divisible
abelian subgroup D which is nontrivial, has finite rank, has finite

index in G, and is characteristic in G.

Proof: (i) Notice the trivial subgroup is a normal divisible

abelian subgroup and hence the collection
45 = {A <1G]A is divisible and abelian]

is nonempty. If 2 is a nonempty totally ordered subset of A and
D = U 2 .9 then D is a normal divisible abelian subgroup of G;
i.e., D e 3 so that D is an upper bound for 1 . Thus by Zorn's
lemma 2 has maximal elements.

(ii) Let G be an infinite Cernikov group, and A a normal

abelian subgroup of finite index with A 6 Min. By 1.22, we write

 

 

If)

u)

LL
1 a-

17

A = A1 +rA + --- + AS for s a positive integer with A1 a group

2
of type p00 (for various primes p) or finite cyclic for each i.
Since A. has finite index and G is infinite, there is some i,
l S_i 5.8 such that A1 is a group of type p”. Let I E_{1,2, ... , s}
be defined by i e I if and only if Ai is a group of type p” for
some prime p. Let D = Z>tAili e I}. [G:D[ = lG:AI[A:D[ < m. Hence
D is a maximal divisible abelian subgroup which is nontrivial, has
finite rank and has finite index.
To see that it is characteristic it suffices to show that it is
the unique minimal subgroup of G of finite index. Suppose H is
any subgroup of finite index in G. Then |D:H lel = IDH:H| < w.
Since D has no proper subgroups of finite index, IlilI)= D; i.e.,
D S_H. Thus D is the unique minimal subgroup of finite index and

as such is characteristic in G.

Notation 2.11: If G is an infinite Cernikov group, then the
subgroup D of 2.10(ii) will be referred to in the sequel as the

minimal subgroup of finite index, and denoted D(G).

When we investigate groups G in the classes L” n I"snb _ a«snb

and L)? n Ifld - "d we will find that the centralizer of a normal
divisible abelian subgroup D plays a significant role. We sharpen

some results of Robinson in the two lemmas below to obtain desired

information.

Lemma 2.12: If D is a normal divisible abelian subgroup of
a group G and H a subgroup of G such that (i) [D,H] £ 1 implies

[D,H,H] < [D,H] and (ii) H/H' is periodic, then [D,H] = 1.

18

Proof: The statement and the proof are only a slight sharpening
of [58; Lemma 5.15]. Let D1 = [D,H] and D2 = [D,H,H]. Define a

mapping T from the set D X H/H’ to Dl/D2 by

(dihH')® = [d,h]D2 .

We claim e is well defined: Let hl, h e H with h H' = h H'.

2 1 2
Then since H' <IH' we have h2hIl e H' and
-1 -1 -l -1
[d,h2] [d,hl] — h2 d ‘hghl dhl
"l "l '
. hl [h2hl ,d]hl e [H ,D] .

But by the 5-subgroup lemma, [H',D] S [D,H,H] = D2. Hence

(d,th')¢ = (d,h2H')¢. Now we claim that Q is bilinear: Let

dl, d2 6 D, h e H. Note that [D,H] is a subgroup of D and hence

centralized by D. Thus

[d

(dld2,hH')¢ h]D2

1d2’

d2
[dl,h] [d2,h]D2

ll

[dl,h1[d2,h1D2

(dl,hH')e(d2,hH')e .

let d.e D and hl’ h2 e H. Then

ll.
Hi

emu.

(d,(th')(h2H'))¢

since is well defined. Now

T

(d,th')o(d,h2H'

Thus (d,(hiH')(h2H'))¢ = (d,th'

h

[d 2,hl]-l[d,hl] e D. Again, si

h
[d2h1'l

’ l [d)h

1]

Thus m

9:D @H/H' —» Dl/D2

is indeed bilinear, and

( '3 denotes

19.

[d, h l2h ]D2
h
2
l] D
h h
2 2

]D

2

h2
[d h2 ][d2

Ml]2’

since D is abelian,

>cp = [d,hglldrhllD2

)T(d,h2H')¢ if and only if

nce D is abelian,

h2hl h2 -l hl

‘1) d d d

(d

h
d 2

h
elm)

-l lh2hl

d

d'1 h2 ] 'lh'l[

h2] l

-l

[d ’h2’hl] e D

2

there exists a unique homomorphism

tensor product) making the diagram

l1 identit H
DXfi-r ______b; D®IF
R4 //:
Dl/D2
comnnlte. [22; Thm. 59.1.] Since T is onto Dl/D2’ so is 9. But
since D is divisible and H/H' is periodic, D(ZDH/H' is trivial.
Hence D1 = D2. By condition (1) we conclude that [D,H] = as desired.

20

Lemma 2.12 allows us to sharpen another result of Robinson [56;
Lemma 2.1(iii)] which states some conditions under which a subgroup H
of a group G will centralize a normal divisible abelian subgroup D

Of Go

Lemma 2.15: If G is a group with a normal divisible abelian
subgroup D and H is any descendant ZD-subgroup of G with H/H'

periodic, then [D,H] = 1.

We postpone the proof of 2.15 until proving 2.15, but we do note

the immediate

Corollary 2.1h: If G is a periodic {Id group and possesses

a normal divisible abelian subgroup D, then D f Z(G).
The following lemma is probably well known.

Lemma 2.15: A group G is a ZD group if and only if for

every nontrivial normal subgroup K of G we have [G,K] < K.

2322:: Sufficiency. We suppose that for every nontrivial normal
subgroup K of G we have [G,K] < K and that 7a(G) £ 1 for all
ordinals d. Then the lower central series must stabilize at some
nontrivial normal subgroup: i.e., there is an ordinal B such that
75(G) £ 1 and. a 2’3 implies 7a(G) = 78(G). But by hypothesis we
(G) = [G, 7

have (G)] < 7B(G), a contradiction as desired.

76+1 5
Necessity. Let G be a ZD-group and

be the lower central series for G. Let K be a nontrivial normal

21

subgroup of G and choose a minimal such that K é_Ga. Suppose

a is a limit ordinal. Then for all B < a we have K E.G by choice

B

of <3 and hence K < (1 <3 == G , a contradiction. Thus a is
-:B<GB a

not a limit ordinal and we have K 5.0a 1 and

1 < x = 0

Since K <IG, we have [G,K] E_K. If [G,K] = K, then K E G3,
contradicting the choice of 0. Hence [G,K] < K and the lemma is

proved.

Proof of 2.15: Let G have a normal divisible abelian subgroup

D and let H be a descendant ZD-subgroup of G, with H/H' periodic.

Case 1. D E_NG(H). Then H'd DH and hence [D,H] is a normal sub-
group of the ZD-group H. By 2.15 we have [D,H] £ 1==#> [D,H,H] <

[D,H]. Hence by 2.12, [D,H] = 1 as desired.

Case 2. D £ NG(H). Then H 7A DH. Since H desc G, we have

DH,? DH,l

H desc DH E_G. Thus H ' < H . Now

HDH,1::}#H): HD

DH,2 z H(HDH’l) = HH[D,H] z H[D,H]

H[D,H]
and

H = H[D,H,H] 0

Hence [D,H,H] < [D,H], and by 2.12 we have [D,H] = 1. But then

D E CG(H) E NC(H), a contradiction. Thus Case 2 cannot occur.

Chapter III

Structure of L" n Iflsnb _ nsnb

In a number of papers, [9-18] and [20], Cernikov studies infinite
groups with particular subgroup properties satisfied by all those
infinite subgroups contained in a specified class of groups. For
example, [11] and [15] study three classes of infinite groups: those
with all infinite subgroups normal, ascendant, or complemented, respec-
tively. In [5A], R. Phillips studies the second of these classes
under the name Ins - "s.

11' X is a subgroup theoretic property, let 1i. denote the class
of groups all of whose subgroups are X—subgroups, and let I? denote
the class of all infinite groups, all of whose infinite subgroups are
I -subgroups, (temporary notation). Belonging to all classes of the
form Ii; would be an infinite nonabelian group with all proper sub-
groups finite, if such a group exists. The question whether such
groups exist has remained open since its formulation by Schmidt in
1958. (See [19] for an expositbry account of Schmidt's problem; also
[58; Sec. 5.h] provides an excellent discussion). Thus, in studying
groups of type Ii’, one must either solve Schmidt's problem or impose
additional restrictions to avoid the problem.

In [11] and [5h] the additional restriction of local finiteness
is placed on the groups studied, which avoids the problem because of
a theorem discovered independently by Kargapolov [28] and by P. Hall
and Kulatilaka [25] (see also [58; Thm. 5.h5]), which says that an
infinite locally finite group always possesses an infinite abelian

subgroup. By using the known results on the class L3 0 Ifls - HS

22

25

we find it more natural to use local nilpotence than local finiteness
in the present discussion. Thus, this paper may be considered an

investigation into the structure of the classes L" n Ifisnb - ”snb

and La n Ind - and.

We will make considerable use of the fact that ITSnb = 37, a

result due to Roseblade, which we state without proof:

Theorem 5.1: There is a function f with domain and range the
positive integers, such that if every subgroup of a group G is sub-
normal with subnormal defect at most s, then G is nilpotent of

class not exceeding f(s).

Reference for proof: [#03 Thm. l] or [59; Thm. 7.h2 and

 

Corollary].
- snb
Corollary 5.2: IT = W.

Corollary 5.5: If G e Irlsnb and H is an infinite normal

subgroup of G, then G/H e "snb = fl .

Notation 5.h: Throughout the remainder of this chapter, f will

denote the function of 5.1.

Lemma 5.5: If G e L" n It! snb and M is a finite normal
subgroup of G with G/M e "r’ then G 6 ’1c’ where c depends only

on r, [M], and the bound b for defects of infinite subgroups.

Proof: Let F be a finite subgroup of G. Then MF is nilpotent

since G 6 L81 and hence

F<l<l MF<l<l G.
I r

[M

21+

That is, every subgroup H of G satisfies H<l<lSG, where

s = maxfb, [M] + r}. By 5.1, we have G e ”f(s).

Lemma 5.6: Let G 6 Lil n Iflsnb. If G has any nonempty

collection H

(“7'7 e I‘} of infinite normal subgroups such that
n“ is finite, then G e "c’ where c depends only on the bound
for the defects of infinite subgroups, the order of DH, and the

class of n H.

Proof: First we suppose that {114 = 1. Let b be the bound
on subnormal defects of infinite subgroups of G. Then for all 7’,
G/H7 e nf(b)' Since ”(H7l7 e T} = 1, we have G isomorphically
contained in the direct product of the G/Hy, which is nilpotent
of class at most f(b). Thus G e nf(b)' The desired conclusion

for the general case, NH ,4 1, now follows from 5.5.

Theorem 5.7: Let G e Iflsnbfifldn L"). If G is nonperiodic,

then G e "snb(fld).

Reference for proof: [5h; Sec. VI; in particular Thm. G].

 

The first major theorem of this chapter is 5.12 below, during
the course of whose proof we need the following four known theorems

which we state without proof.

Theorem 5.8: Let G be an abelian by finite p-group. Then

G is a Cernikov group if and only if its center satisfies Min.

Reference for proof: [7; Thm. 5] or [59; Lemma 10.21].

 

[0
\J‘l

Theorem 5.9: The locally nilpotent groups which satisfy the
minimal condition on normal subgroups are precisely the hypercentral

Cernikov groups.

Reference for proof: [6], [52], or [58; Thm. 5.27, Corollary 2].

 

Theorem 5.10: A minimal normal subgroup of a locally nilpotent

group G is contained in the center of G and is of prime order.

Reference for proof: [52].

 

Theorem 5.11: A periodic hypercentral group is a Cernikov group
if and only if each upper central factor with finite ordinal type

satisfies Min.

Reference for proof: [553 Thm. 8] or [59; Thm. 10.25, Corollary 1].

 

The following theorem obtains (L31 n Iflsnb - :1) 3 Min and

shows that we may limit our attention to p-groups in (L)? fl I?! snb - fl ).

Theorem 5.12: G 6 Ln fl Iflsnb

- 11 if and only if G = P + K,
where P is a Sylow p-subgroup of G, P is an infinite Cernikov

group, P 6 L11 fl Iflsnb - fl and K is a finite nilpotent group.
Proof: Necessity. Let G e Lt! n Insnb - :1. Let
A = (l {H <1 GlH is infinite} .

By 5.6, .A is infinite. Since A e L3! and since by 5.7, A is
periodic, we may write A = 2‘, [Aili e I}, each A1 a Sylow pi-subgroup
of .A. Notice that Ai char A; thus Ai <lG- for all i e I. If

[I] >'l, A1 is finite for all i e I since A can have no proper

26

infinite subgroup which is normal in G. Hence [I] >>1 implies

that I is infinite since A is infinite; but then A has a proper
infinite subgroup which is normal in G. Hence [I] = 1; i.e., A

is a p-group. Let P be a Sylow p-subgroup of G containing A.

Since G e L]! and periodic write G = P + K, where K is a Sylow
p'-group of G. If K is infinite, then by definition of A and

P we have A 5 PK) K = 1. Hence K is finite and K T_.' G/P efl by
5.5. Clearly P ,é 1'] since G = P + K £1! . However, P e LII/l If! snb
and hence it remains only to show that P is a Cernikov p-group. To
this end we claim first that A' is finite. Suppose A' is infinite.
Since A' char A char G, A' <1 G and hence A' = A. Since P e L“ E Z,
we have for all x, 1 £ x e A, [P, xP] < xP. Hence xP < A. Since

A has no proper infinite subgroups which are normal in P, xP is
finite. Thus P/CP(xP) is finite and hence A E CP(xP). Since x e.A
was arbitrary we have A' = 1, a contradiction. Thus A' is a finite

normal subgroup of P.

Next we claim that A is a hypercentral Cernikov group. Let

be the maximal elementary abelian subgroup of A/A'. Suppose that
B/A' is infinite. Then A S B; i.e., A = B and A/A' is infinite
elementary abelian. Let F i P be finite. P/A e us’ for some s.
Thus

(1) [P, F, , F] g [P, AF, , AF] _<_A .
(---s----§ (r---S ----- )

Now AF/A' is abelian by finite and is not a Cernikov group. Thus

27

by 5.8 its center is infinite. Hence Z(%) n £1- 18 also infinite,
and an elementary abelian p-group; i.e., it is an infinite direct
sum of cyclic groups of order p and contained in the center of AF/A'. 5

snb

Thus AF/A' e L” fl I31 (bound b) and has an infinite descending

chain of normal subgroups with trivial intersection. By 5.6, AF/A' e "c’

for some c. Thus

[P, F, ... , F] < [A, AF, ... , AF] <.A'
€--S+c---? — (“"c-"u9 -

where we have used (1). Hence every subgroup of P/A' is subnormal
with defect at most maxfb, s + c}. Thus P/A' e 1". By 5.5, P e N,
a contradiction. Thus B/A' is finite; i.e., A/A' has Min and
so A 6 Min. By 5.9, A is hypercentral Cernikov p-group.

Next we claim that P e ZA. Since A <1 P and A 6 Min we
may choose 1 ,4 B E A, B minimal with respect to B Q P. By 5.10,
Bgz(P). Hence Z(P) ,4 1. Now P/Z(P) 6 Ln!) stnb - w and

so by induction
Zn(P) < Zn+l(P)’ n = O, l, 2, 000

whence Za)(P) = U Zn(P) is infinite. Thus P/Zw(P) e If by 5.5
n < as
and we have P e ZA.

Next note that Zn(P) is finite for all n < a) since otherwise
P/Zn(P) is nilpotent for some n, implying that P is nilpotent.
Thus by 5.11, P is a Cernikov group. This completes the necessity.

Sufficiency. Let G = P + K, P an infinite p-group,

P e L“ n Iflsnb - Y! and K a finite nilpotent p'-group. Then

G e L" - fl and we claim furthermore that G e Insnb. Let r be

28

the bound on subnormal defects of infinite subgroups of P and K 6 31¢

Let H be an infinite subgroup of G. Since H e LII, write

H = + H where H is the S low ~sub ou f H and H is
Hp Pt: p Y P gr P 0 P'
the Sylow p'-subgroup of H. Then Hp E_P and Hp, < K; _hence HP

is infinite. But then we have
H = H + H ,'d'< 1H + K did P + K ,
P P C P r

whence every infinite subgroup of G is subnormal of defect no more
than c + r, as desired.

For the next theorem about the structure of p-groups in the
class L" n stnb - if, we need the following lemma which we state

without proof:

Lemma 5.15: Let D be a divisible abelian subgroup and F a

finite subgroup of a group. If DF = D, then D = CD(F)[D,F].

Reference for proof: [58; Lemma 5.29.1].

 

Now we restrict our attention to p-groups in the class
L11 0 Iflsnb - fl . The next theorem gives a version of their structure

which is further pursued, together with examples, in Chapter VI.

Theorem 5.1%: Let P be an infinite p-group. Then
P e L?! n Ifisnb - if if and only if P is a Cernikov p-group satis-
fying the following: Let D = D(P) and c = CP(D). (i) c < P with
C e I! , and (ii) x e P - C implies x does not normalize any infinite

proper subgroup of D.

29

M: Necessity. P is Cernikov by 5.12; by 2.lO(ii), D is
a divisible abelian p-group of finite index in P. Thus D S Z(C) and
hence C/Z(C) is a finite p-group and thus nilpotent. Thus C e 6‘]
and we conclude that C < P. Now let x e P normalize an infinite
proper subgroup H of D. Then D = CD(x)[D, < x >] by 5.15. If
CD(x) is finite, then [D, < x >] has finite index in D. Since D
has no proper subgroups of finite index, we have for all n 2 l,

[D, <x>, <x>, ,<x>]=D. But, H<lD<x> and H is
(- -------- n-u-uh ------ --’
infinite. Thus by hypothesis and using 5.5

D<x>
T6"-

Hence there exists an r such that

[D,<x>,<x>, ... ,<x>]f_H<D,
l

——r------------)

 

‘

a contradiction. Thus we have CD(X) infinite. Thus
< x > q CP(x) <1 <1s P, where s is the bound on subnormal defects of
infinite subgroups of P, and by 2.15 we have [D,x] = l,- i.e.,
x e C = CR(D).

Sufficiency. Let H be an infinite subgroup of P. We claim
that either D E H or H E C. For suppose otherwise. Then H n D
is a proper infinite subgroup of D and there is some x e H - C.
By condition (ii) we have x £ NP(Hfl D). Bit H n D 4 H and. x e H,
a contradiction, which establishes the claim. But now clearly P e Iflsnb
because D _<_ H implies H 44d P where P/D e ”d (P/D being a finite
p-group) and H S C implies H <l<lc C <1 P, where C e fie. Hence

H<l<ls P, where s=max{d,c+l}. Now let xeP-C (we use

5O

condition (i)). By 2.13 We have < x > is not descendant in P.
Thus P i if . A Cernikov p-group is locally finite and hence in L37

and so P eLfln Iflsnb - 7!.

It is primarily because of [5’4] that we have restricted our
attention to locally nilpotent groups instead of locally finite groups.

We now consider the relationship of 5.11; to [515' Thm. D].
Lemma 5.15: ([5h; Lemma 5.1]) {(50 L3 = Lfln L1.
Lemma 5.16: (Lin Iflsnb - '7) - (Lyn Ins - its) =

mm mm - n.

snb

Proof: Clearly L3 n In 5 L! n Ins. Therefore

(La n Insnb - rt) - (L! n Ins - as) La nmsnbnns - rt

L3 nLun Insnb - t!

where we have also used 5.15. But G e L" n Iflsnb - fl implies
G is a Cernikov group (5.12) and hence G (5 L3. The desired result

follows .

It is an unsettled question whether (L3 I) If - 373) ..

(L! n Iflsnb - fl) is empty; equivalently whether L! n Ins .. n3

_<_ Iflsnb. To answer affirmatively, it suffices to show that

L3 n Il‘ls - :15 _<_ Ind because L! n Ins - as 5 Min (see 5.17
below) and it is known that if c- satisfies Min, then there is a
bound on the subnormal defects of subnormal subgroups (see [583

snb

Corollary to Thm. 5.h9]). Therefore Il‘ldfl Min _<_ Iflsnfl Min 5 If!

Another equivalent formulation of this question is whether a group in

31

the class L? n 1318 - "3 must satisfy the conditions (i) - (iii)

of Theorem 5.19 below. But first some preliminaries.

Theorem 5.17: ([5h; Theorem D]) G e L! n Ifls - ”s if and

only if G has a normal Sylow p-subgroup P satisfying

(1) P is a Cernikov p-group,
(ii) G = P]K where G/D(P) is a finite nilpotent group,

(iii) If x e K normalizes an infinite subgroup of D(P), then

x e C(P), and

(iv) There is an x e K such that [x,P] ,4 1.

M118: (Du, 2.h.l and 2.h.2]) (i) If G e L? and P is
a normal Sylow p-subgroup of G with G/P countable, then P has
a complement. (ii) If G 6 L3 and locally solvable and G = P]K
where P is a Sylow p-subgroup of G and K is finite, then any

Sylow p'-subgroup L of G is conjugate to K.

Theorem 5.19: G 6 L3 (7 stnb - W if and only if G has a
normal Sylow p-subgroup P such that P is an infinite Cernikov group
and G satisfies either conditions (i) - (iii) or conditions (1') -
(111') depending on whether G £ Lfl or G 6 L17, respectively.

Let D = D(P) and c = CP(D).

(i) G=P]K,G/De 0103 (i') G=P+K,G/Defl03

(ii) CG(D) e n and K sér G (11') c < P with c e W

52

(iii) If x e G normalizes an (iii') x e P - C implies x does
infinite proper subgroup not normalize any infinite
of D, then x e CG(D)° proper subgroup of D.

If, in addition, x e K,

then x e CG(P).

2322:: Sufficiency. Let G satisfy (1) - (iii). Clearly
G 5 L3, and since K sér G, we have C £ W. We need only show
that G e Iltsnb. Notice that G is locally solvable since every
finitely generated subgroup is an extension of a finite p-group by
a finite nilpotent group, hence is solvable. Let H be an infinite
subgroup of G. Then H n P is an infinite normal Sylow p-subgroup
of H and by 5.18 we may write

H=(HflP)]L,

where L is a p'-group, and for some t e G, Lt E.K. NOw

L S NG(D) n NG(H n P)
and hence
(*) H n P fl D = H n D is an infinite subgroup of D ,
normalized by L.
Case 1. H n D = D. Then D S H and hence
E

G
D‘Q‘4-— e 11

giving H <3 <1 G.

35

Case 2. H n D < D. We claim that L E CG(P). On account of (*)
and condition (iii) we have that L E CG(D). Let x e L. Then xt
is an element of K which centralizes the infinite proper subgroup
of D, H nD. By the second part of condition (iii), xt e CG(P)

and hence x e CG(P). Thus L _<_ CG(P). Hence, H = (H (I P) + L.

Now we claim that H fl P 4 <1 P. We assume without loss of generality
that P A n . Then G = CP(D) e )1 since 0 _<_ CG(D) e I! by con-
dition (ii). Thus C < P. Furthermore, x e P - C implies that

x does not normalize any infinite proper subgroup of D by condition
(iii). Thus, P satisfies conditions (i) and (ii) of 5.1h, so that

PeIflsnb. Hence, HOPQQP, and we have
H=(HnP)+L<1<1P+L<l<IG,

where the last relation follows from the fact that G/P is nilpotent,
being an image of G/D.

Thus, in both Case 1 and Case 2 we have H Q Q G, and we have
shown G s It! snb. Hence also G 6 L3 0 Iflsnb - 7!.

Now we show G A L”. Since G satisfies conditions (i) - (iii),
C also satisfies conditions (i) - (iii) of 5.17. If [K,P] = 1, then
G = P + K, and K <1 G, violating (ii). Hence G satisfies (iv)
of 5.17 as well, showing G e L3 n 1318 - 818. By 5.16, G s L”.

That conditions (i') - (iii') imply G e L?! n Insnb - 8'!
follows from 5.12 and 5.1h. Furthermore, G 6 L3 , since G is a
Cernikov group. This completes sufficiency.

Necessity. Let G e L, n Iflsnb - )1. If G e L“, we are
done, by 3.12 and 3.11;. Suppose G £ Ll'l. By 3.16, G e L! n 1318 - :18.
Hence, by 5.17, G has a normal Sylow p-subgroup P which is a Cernikov

group with D = D(P), C = CP(D) and furthermore, G = P]K, G/D e I! n 3 .

5:.

We claim that condition (ii) is satisfied. Now D E Z(CG(D)) and

_ CG(D) . _
hence C = ETEETETV is a subgroup of an image of G/D. Hence C
is nilpotent, giving CG(D) e '1. Since a serial Sylow p'-subgroup
of any group is normal, K sér G. Thus we have (ii).

To show (iii), let x e G normalize an infinite proper subgroup
H of D. ‘We argue exactly as in 5.1h that CD(x) is infinite. Then

since G e I'Isnb, we have

<x> <1 CG(x) <I<IG

and by 2.15, [D,x] = 1; i.e., x e CG(D)' If x e K as well, then

by 5.17(iii) we have x e CG(P). Thus we have condition (iii).

Chapter IV

Structure of Lt! n 131“ .. ryd

In this section we present a structure theorem for L” n Ifld- if
analogous to 5.12 and 5.1%. It is still unsettled, however, whether
such groups satisfy Min. If they do, then Lflfl II'Id - I'Id =
L“ n Iflsnb - W. Constructing an example without Min seems to be
a difficult problemuthe Heineken-Mohamed group seems invited into
the construction, but it cannot be used in the obvious place, as we
shall see. Since the study of L)? n stnb - 1'! made considerable
use of the fact that "snb = I", it may be that the question will
only be settled after significant information is obtained about p-groups,
all of whose subgroups are descendant, not satisfying Min, and

possessing a finite non-trivial abelian image (sec 14.7).

Lamina L1: If G 6 LI! ”Ind and M<l G, M a finite sub-

group with G/M e 316‘, then G 6 11d.
£12932: Let F be finite. Then MP is nilpotent and hence
F G <1 MF desc G .
Thus F desc G.

Lemma h.2: Let G 6 LI! n Ifld. If G has any nonempty collec-
tion C = {H7l7 e 1‘ I of infinite normal subgroups such that DC is

finite, then G e fld.

Proof: By 2.5 we assume without loss of generality either If] = 2

or C is a descending chain. By h.1 we further assume that fit = l.

35

56

Case l. [P] = 2. Let C = {H,K}. Then H, K <1 G, PM K = 1. Let
F be a finite subgroup of G. By 2.5., we have IF n K!" finite,
hence nilpotent. Thus F <l< HF fl KI" desc G since the intersection

of descendant subgroups is descendant.

Case 2. C is a descending chain, say C = {Hula < G}. Let B
be an ordinal such that for all infinite subgroups H of G, we

have HG’B S H. Let F be a finite subgroup of G. Then

(naNG’B 5 Ha? ,

for all a < 0. Thus FG’B _<_ n (Hal?) = F where the last equality
a < 0'

comes fran 2.2.

The following theorem shows we may limit our attention to p-

groups in the class Lfln Ind - I'd.

Theorem 14.5: G e L” n Ind - I'd if and only if G = P + K,
where P is a p-group in the class Lflfl Ind - nd and K is

a finite nilpotent p'-group.

35333;: Necessity. Let A = n [H <1 GIH is infinite}. By h.2
A is infinite. We prove exactly as in 5.12 that A is a p-group
and G = P + K, where P is a Sylow p-subgroup of G and K a
Sylow p'-subgroup of G. Furthermore K is finite. That
P e L” n Ifid is clear. Suppose P 6 ad. Let H be any subgroup
of G. Since G is periodic (5.7) and locally nilpotent, elements

of relative prime order commute, and we write

H=(HnP)+L, LtEK, forsome teG.

37

Now by supposition, HnPdesc P and G/PgKefl and hence
H=(HnP)+LdescP+L<1<IP+K=G.

Thus G 6 ad, a contradiction, assuring that P e L)? n Ind - fld.
Sufficiency. Let G = P + K be as in the statement. Clearly
G e L)? - fld. We claim G e Ifld as well. Let H be an infinite

subgroup of G. As above, write

H = (H n P) + L, Where [P’L] = l, I:t _<- K n

Then L is finite and hence H n P is infinite. Thus
H= (HnP) +LdescP+L<l<lP+K=G.

Theorem lull: Let P be a locally nilpotent p-group. Then
P e Ind - fld if and only if P has a normal divisible abelian

subgroup D of finite rank with c = CP(D) satisfying

(1) P/D e «d.
(ii) 0 < P and c e nd,

(iii) x e P - C implies x does not normalize any proper infinite

subgroup of D, and

(iv) if H is an infinite subgroup of P, then either H _<_ C

or D_<_H.

Proof: Necessity. Let A = n (H <IPIH is infinite). By l+.2,
A is infinite and by construction contains no proper infinite subgroups
which are normal in P. We argue exactly as in 5.12 that A' is a

finite normal subgroup of P. Let TIE-r = fi-fip) be the maximal elementary

58

abelian subgroup of A/A'. Suppose that B is infinite. Then A = B;
i.e. , A/A' is an infinite elementary abelian p-group. Let F be
a finite subgroup of P. Then AF/A' is abelian by finite and not

a Cernikov group, so it has infinite center by 5.8. Thus
AF A
Z(Kr) n it

is also infinite. Thus AF/A' e LU! fl Ifid and has an infinite
descending chain of normal subgroups with trivial intersection. By
1+.2, AF/A' e Nd and by h.l, AF 6 I'd. Hence F desc AF desc P;
but then P e fld, a contradiction. Thus B is finite and A/A' e Min
and hence A e Min, meaning that A is a hypercentral Cernikov group
(5.9).

Now let D be a maximal normal divisible abelian subgroup of
P (use 2.10(i)). Clearly D is nontrivial. If rank D is infinite,
then D(p) is infinite and normal in P; hence A S D(P), a con-
tradiction. Thus D has finite rank. Let C = CP(D). Condition
(i) is satisfied since D is infinite and P e Ifld. Now C e fld,
because any group in It‘ld with infinite center is in 81d. (Let
F be any subgroup of C. Then F d Z(C)F desc C.) Condition (ii)
is now apparent.

Now let x e P normalize an infinite proper subgroup H of D.
Suppose CD(x) is finite. Since D = CD(x)[D, < x >] by 5.15, we
have [D, < x >] a subgroup of D of finite index; thus [D, < x >]

= D. But then

D<x>

(H<x>) =H<x>[D<x>,H<x>]ZH<x>D=D<x>5

1.8., H < x > dis/so D < x > so that D < x > i Ind. The contradiction

59

assures that CD(x) is infinite. Then
< x > <1 CP(x) desc P

and by 2.15 we have [D,x] = 1; i.e., x e C. Thus we have condition
(iii) as well. To obtain condition (iv), let H be any infinite
subgroup of P and assume that H £_C. Let x e H - C. If CD(x)

is infinite, then x centralizes an infinite subgroup of D so that
by condition (iii) it must centralize D, contradicting x é C.

Thus CD(x) is finite and so by 5.15 [D, < x >] is a subgroup of

D of finite index. Hence [D, < x >] = D. NOw

HDH = H[DH,H] .>_ H[D, < x >1 = HD = DH .

But since H desc DH, we have H = DH, whence D S_H, as desired.
Sufficiency. Let H be an infinite subgroup of P. If H E_C,
then H desc C <1 P and hence H desc P. Otherwise D S H by con-

dition (iv) and

H P
E desc B

by condition (1), and hence H desc P. Thus P e It'd. By condition
(ii) there is some x e P - C; by condition (iii) and by 2.15

< x > ddsc P. Thus P l fid-

Because of the similarity of h.h with 5.1h, the question arises
whether conditions (i) - (iii) in h.h actually imply condition (iv).
The following example for p = 2 shows that this is not the case.

In fact the construction can be carried out for arbitrary primes p,

fbr one only needs to know the existence of a divisible abelian p-group

’+O

of finite rank and an automorphism of order p which satisfies con-

dition (iii). We show the existence of these in Chapter V.

Example h.5: Let D be a 200 group, E an infinite elementary
abelain 2-group, 9:E -’ < a > where a e Aut D is the automorphism
sending every element to its inverse. Let P = D16 E. Let K = Ker 9.
Then CP(D) = D]K = D + K is abelian, hence in 116‘, P/D ; E is
also in ”(1, and x e P - C implies that x normalizes no proper
infinite subgroup of D. Yet if P e Ifld, then P 6 ad because
K 5 Z(P), and K is infinite elementary abelian so that P has an

infinite descending chain of normal subgroups intersecting in 1.

We next collect miscellaneous information about the groups in
h.’+, some of which could be helpful in deciding whether such groups

must satisfy Min. First, we make the following

Definition l+.6: A group G is a Heineken-Mohamed group if G

 

is a metabelian p-group with the following properties: (i) G' is
elementary abelian, (ii) every proper subgroup of G is subnormal and
nilpotent, and (iii) Z(G) = 1. Examples of such groups were discovered
by Heineken and Mohamed [27]. It is known [27; Lemma 1] that such

a group G also satisfies (iv) G/G' is a group of type pm.

Lemma 1$.7: Let P be a p-group, Pe er Ifld - fld. Let
D and C be as in ink. Then the following additional properties
hold:

(1) for all n _>_ O, Zn(P) < Zn+1(P) and Zn(P) is finite.

(ii) z(c) eMin and P/C is finite.

#1

(iii) PeMin ifandonly if Ce nnMin ifandonlyif

c/z(c) is finite.
(iv) P has a local system of nondescendant subgroups.

(v) P/D has a finite nontrivial abelian image; thus P/D is

not a Heineken-Mohamed group.

(vi) if P A Min, then D has no proper supplement; thus in

particular, P is not a split extension of D.

The proofs of portions of the above lemma require the following

three results due to Baer.

Theorem h.8: G e finMin if and only if Z(G) e Min and

G/Z(G) is a finite nilpotent group.

References for proof: [1; Sec. 6, Satz 2] or [58; Theorem 5.1h].

 

Lemma h.9: Let D be a divisible abelian p-group of finite
rank, and (I an automorphism of D which fixes every element of
order p and, when p = 2, every element of order A. Then (1 has

infinite order.
References for proof: [2; p. 525] or [58; Lemma 5.28].

Corollary h.10: Let D be a divisible abelian p-group of finite

rank. Then periodic subgroups of Aut D are finite.

Reference for proof: [58; Corollary to Lemma 5.28].

 

Proof 9_f_ l$.7: (i) Since D 6 Min and D <1 P, we may choose B,

1% 3 <1 D, minimal with respect to 3 <1 P. By 3.10, l ,é B 5 Z(P).

1+2

Thus ZO(P) < Zl(P). If Z1(P) is infinite, then for all finite
subgroups. F of P, we have F <1 Zl(P)F desc P; hence P 6 Ha.

Thus Zl(P) is finite. suppose for induction that zn l(P) < Zn(P)

and Zn(P) is finite. Then by n.1, we have P/Zn(P) e Ln n Ind .. nd

and so by the case already established, P/Zn(P) has finite nontrivial

center. Thus Zn(P) <Z (P) and Z (P) is finite.

n+1

(ii) As in the proof of u.u, let A be the intersection of all

n+1

infinite normal subgroups of P. A is infinite and A e Min. If
Z(C) ,é Min then [Z(C)](p) is an infinite normal subgroup of P.
Thus A f [Z(C)](p), a contradiction. Thus Z(C) 6 Min. Now P/C
is isomorphic to a periodic subgroup of Aut D and hence is finite
by h.10.

(iii) First we show P 6 Min implies C e an Min. By 1+.14
we clearly have P 6 Min implies C e «an Min. But fldn Min
5 "sun Min is clear, and since a group satisfying Min has a bound
on the subnormal defects of its subnormal subgroups [58; Corollary to
5.19] we have “snn Min E ”snb = W. Next we note that C e H n Min
implies C/Z(C) is, finite, by using (ii) and h.8. Finally, we show
that c/z(c) finite implies P e Min. Since 2(0) 6 Min by (ii),
we have C 6 Min, and since P/C is finite also by (ii), we have
P e Min.

(iv) This fact was pointed out to me by my advisor. Let F
be a nondescendant subgroup of P, and L any finitely generated
subgroup. If < F,L > desc P, than F <1 <l< F,L > desc P, since
< F,L > e )7 because F is necessarily finite. Hence < F,L > da’sc P.
Hence {< F,L >|L is finitely generated, L E P} is a local system

of nondescendant subgroups of P.

1+5

(v) Since D 5.0) P/C is a homomorphic image of P/D. Thus
it suffices to show that P/C has nontrivial abelian image. But P/C
is a finite nontrivial p-group, and hence is nilpotent. Let F = P/C.
Then F' < F and F/F' is a finite nontrivial abelian image of P/C.

Now let H be a Heineken-Mbhamed group. Suppose H has a
finite nontrivial abelian image. Then H/H' has a proper subgroup
of finite index. But H/H' is a grou; of type p” by h.6(iv), which
has no such subgroups.

(vi) Suppose P é Min, and let H be any supplement of D;
i.e., let P = DH. If lP:Dl is finite, then P e Min. Hence H
is infinite and by 1+.’+(iv) we have either H E C or D S H. But

His C implies DH E_C < P. Thus D E’H and we have P = DH = H.

We remarked at the beginning of this chapter that deciding the
question of whether the groups studied in this chapter satisfy Min
might depend on knowing the structure of infinite groups, all of whose
subgroups are descendant, not satisfying Min, and which have a finite
nontrivial abelian image. For if P i Min, then P/D is such a

group. NOW we list some elementary results related to these groups.
Lemma h.11: (i) Let G be a periodic ’Td" group. If G has
a normal divisible abelian subgroup D, then D E_Z(G).
(ii) ndnmnemnmn: rm Min
. d . . HK
(iii) G e f! if and only lf for every H < Kif G we have < K.

(iv) znf nd and Lnnfldgzn.

hi4

Pr_oo_f: (i) Let l ,4 x e G. Then < x > is a descendant abelian
subgroup which is periodic. Thus by 2.15, we have [D,x] = 1; hence
D E Z(G).

(ii) fin Min_<_ZDn Minnrrd is clear. ZDnMin_<_ an Min
is also clear since the lower central series must stop after finitely
many steps. All that remains is war) Min 5 n . But this fact we
already noted in the proof of h.7(iii).

(iii) This is obvious.

(iv) Suppose ZD 5 81d. Let F be a free group. Then F is
residually a finite p-group; i.e., F is a ZD group since all resid-
ually nilpotent groups are ZD groups. By hypothesis every subgroup
of F is descendant. If G is a finite image of G, then G is

nilpotent; i.e. , every finite group is nilpotent. Thus ZD f 876‘.

Let G be the Heineken-Mohamed group displayed by Hartley in

[26] as a subgroup of the restricted wreath product W of a group

.. ._ P _ P _.
C _ < c > of order p by a group U — (ul, u2, ...lul .. l, ui+l _. ui,
i = l, 2, ...) of type pm. The base group B of W is naturally

a right module for the group ring ZPU. Hartley obtains certain
u.-1

elements ai e B l and sets Zi = uiai, (i = 2, 5, ...). Then

G = < 22, z > is the Heineken-Mohamed group. Every Heineken-

}, coo
Mohamed group is an extension of an elementary abelian p-group by a

p-group and hence is locally a finite p-group. Furthermore every
subgroup is subnormal. Thus G 6 Ln n rid. We claim that G é ZD.

u2-l u2 -1
- c

Let a = c _ c . Now from the equation two lines below (5)

of [26], we have

#5

T (ui+1-l)P-l ] [ (uisrl"l)]p-li‘ul
is , 2m] 2w J
(*) = [zi+l) zi]
u.-1
l
= a .

.A
From equation (5) of [26] G' = a , where A is the augmentation
ideal of Zéu. Hence the left-hand side of (*) is in 73(G) = [G,G,G]
and hence for all i = l, 2, , we have

u.-l
a l e 75(G) .

But 73(G) is a ZPU submodule of B (since G' is a EPU submodule
of B) so that

ui-l

a 675(G) for all 1 1,2, ... , and OEm<p1.

Thus 5A = G' = 73(G). NOW G'

ll

72(G) is nontrivial (in fact infinite)
and hence G é ZD.
We make one final remark regarding the question of whether

Lfln Ifld - fldEMin.

Remark L12: If L“ n Ind - "C1 E Min, then L“ n Ind - "d

= L“ n Insnb ' ”0

Proof: Clearly, Ifldn Min 5 Iflsnfl Min. Since a group
satisfying Min has a bound on the subnormal defects of its subnormal
subgroups [58; Corollary to 5.19] we have Iflsnn Min 5 Iflsnb, and

the desired result follows.

Chapter V

Strongly Irreducible Antomorphisms

If P is an infinite p-group and either P 6 LI! n Iflsnb - Y]
or P e L“ n Ind - fld, then P has a normal divisible abelian
subgroup D of finite rank, with C = CP(D) < P and P/C a finite
group of automorphisms of D with the property that nontrivial elements
of P/C normalize no proper infinite subgroup of D (5.1M and h.h).
Further investigation of such automorphisms in this chapter leads to
some interesting relationships between the classes studied in Chapters
III and IV, and direct limits of p-groups of maximal class. Those

relationships will be investigated in Chapter VI.

Notation 5.1: Throughout this chapter, p will denote a fixed

prime and D a divisible abelian p-group of finite rank, r.

Lemma 5.2: Let D = D1 + D2 + ... + Dr’ where Di has gener-

ators x and defining relations pxi l = O, px
)

. X. ... 0
1,1’ 1,2’ l,n+1

=X.

1 n’ n = l: 2: ... . Then every a e D, a £ 0, has unique repre-
’

sentation in the form

* a = m X + m x + -~- + m x
( ) 1 1,n 2 2,n r r,n ’

. n
where Ia] = pn and the m1 are integers, O f.mi < p ,

i = 1, 2, ... , r. Equivalently, multiplying as matrices we may write

... X ) .

** = ...
( ) a (m1 m2 mr)COl(xl,n x2,n r,n

Proof: It is clear that a £ 0 has a representation (*).

Suppose

1+6

LG

: + so.
a Sle,t S2X2,t + + err,t

is also of the form (*). Then lal = pn = pt. Hence n = t. Further-

more, since D is a direct sum of the Di’ we have

m.x. = S.X. for all i = l, 2, ... , r .
l 1,n i 1,n
ki J.i
. = ,8 = = 2
Write mi p Bi, ( i’ p) 1 and 3i p ti, (ti, p) 1. Then
ki
mi§1,n = P Elxl,n — i 1,n-ki
and
Ji
8 x - p t x — . . . .
Now
mixi,nl = lsixi,n| implies n - ki = n - Ji
implies k1 = Ji .
Furthermore,
k.
i
= < .
mi _ si(p ) and k1 n
Thus
n

and since 0 E_mi < pn and O 5.81 < pm, we have m1 = Si'

As we shall soon see, the above representation lends itself well
to describing the action of automorphisms of D by use of r X r
matrices over the p-adic integers. Our following description of the
p-adic integers is taken from [5; pp. 20 ff.] except that we write

our sequences on the set of positive integers instead of on the set

of nonnegative integers.

 

A8

Definition 5.5: A sequence of integers {xn} satisfying
xn+l 5 xn(pn) for all n 2’1 determines a p-adic integer, where
two such sequences [xn} and {x5} determine the same p-adic integer
if and only if xn E xr'l(pn), for all n 3 l. The set of p-adic

integers, denoted by RP, form a commutative ring under componentwise

addition and multiplication.
Lemma 5.h: Let a 6 RP. Then

(i) a has a unique representation as a sequence of integers

n n
<
{Kn} such that Xn+l _ xn(p ) and O E-Xn p for

all n 2.1.

(ii) If (1 = {xn} is the representation of (i), then a is a

unit of Rp if and only if xl # 0.

(iii) If’ a £ 0, then a has a unique representation of the
form a = pmg, where m is a nonnegative integer and

g is a unit. (See Lemma 5.5(ii).)

Proof: (i) is obtained simply by reducing the representation
of 5.5 modulo pn. (ii) and (iii) are [5; Thm. 1] and [5; Thm. 2]

respectively.
Lemma 5.5:
(i) RP is a PID and hence also a UFD.

(ii) a? = [E's §](n,p) = 1} is a subring of RP. In particular

Z S RP. Positive integers have representation 5.h(i)

as eventually constant sequences and negative integers have

A9
representations computed from

T1

-1 = {P - l} .

Proof: (i) follows from [5; Corollary 2 of Thm. 2] and 5.4(iii).

(ii) is [5; Corollary 1 of Thm. 1].

Definition 5.6: Denote by FP the quotient field of RP. For
q a prime (not necessarily distinct from p), denote by ¢q the

(cyclotomic) polynomial with coefficients in RP given by

¢q=Xq-1+Xq-2+ooo+x+l.

meaSJh

(1) Every' o e Fp, a,£ O, has unique representation a = pmg,

m an integer, g a unit of RP.
(ii) The characteristic of Fp is zero.
(iii) ¢p is irreducible over Fp.

(iv) If q # P is prime and ¢q factors into relatively prime
polynomials f and h of degrees m and n respec-
tively over 2?, then ¢q factors into polynomials of

degrees m and n over RP.

22223: (i) is [55 Thm. h].

(ii) The quotient field of a domain always has characteristic
zero.

(iii) It is easy to show that over any domain R, f e R[X] (the
,polynomial ring over R) is irreducible if and only if f(X+l) is

irreducible. Then ¢p(X+l) is irreducible by Eisenstein's criterion:

(x+l)p - l
(X+l) - l

 

¢P(X+l) =

(iv) Since Rp/(p)

[“25 P- 279])-

- Xp'l

5?,

50

p-2 + P(P-l) XP-5

+ pX 1-2

+ooo+p.

this follows from Hensel's lemma (see

Notation 5.8: Let GL(r, RP) be the ring of all r X r matrices

over RP with determinant

 

Definition 5.9: For

as follows:

Let a =
a.. =

13
d _

where
Then

‘P

(dwarf) = (m

 

)

a unit of R .
P

every a e GL(r, RP), define <1®zD * D

(aij) )
ij m
{bIn )m=l e RP ,
x1,n
x
2,n
(m1 m2 .. mr) . ,
x
r,n
n
ldl = P -
X1,n
x
2,n
1 m2 mr) . [(aij)¢]
x

x1,n
.. x
_ ... lJ 2,n
_ (m1 m2 mr)(bn ) . ’
x
r,n

51

where (ml °°° mr)(b:3) denotes usual matrix multiplication.

”‘2

00
Lemma 5.10: (i) The endomorphism ring of a group of type p
is isomorphic to RP. (ii) For every a e GL(r, RP), am e Aut D.

(iii) T is a ring isomorphism of GL(r, RP) onto Aut D.

References for proof: [21; Thm. 55.1] or [50; pp. l5h-l57].

 

Remark 5.11: In view of 5.10(i), D is a left module over the
PID, RP, in a natural way. Unfortunately D is not finitely generated
as an Rp-module; for if X is any finite subset of D, then there
exists a positive integer n for which X is contained in D(pn),
and D(pn) is clearly a proper submodule of D. Furthermore, D is
not a vector space over Fp; for suppose it is. Using the notation

of 5.2, for p.1 e Fp we have

Thus p-1 is not an endomorphism of D since it carries an element

of order p into an element of order p2.
Following [5h], we make the following

Definition 5.12: Let ate Aut D. We call. a a strongly irre-

 

 

ducible automorphism (abbreviated S-I automorphism) if no proper infinite

 

subgroup of D is a-invariant; i.e., if for all H < D, H infinite,

we have

52

Notice that elements of P-C in 5.1h and h.h are essentially S-I

automorphisms of order a power of p.

Lemma 5.15: let a e Aut D. Then a is an 8-1 automorphism

if and only if for every proper nontrivial divisible subgroup D1 of

D we have
(1

Proof: Sufficiency. Suppose a is not an S-I automorphism.

<
By 5.12, choose H < D, H infinite such that H a > < D. Let Dl
. . . . <Ia >
be a max1ma1 lelSible subgroup of H . Note Dl < D and D1 £ 1
<
since H is infinite. Furthermore Dl char H a > and hence
Ct
Dl — Dl .

Necessity. Suppose there exists a proper nontrivial divisible

subgroup D of D such that

1
oz
D1=Dlo
an
If D1 = Dl for some positive integer n, then
n+1 n

O! (10! or

D1 =(D1)=D1=Dl°

<la >

D < D. Hence a is not an S-I auto-

Hence, by induction D1 1

morphism of D.

Lemma 5.1h: If a is an 8-1 automorphism and 8 any automorphism

B

of D, then a is an 8-1 automorphism of D.

55

Proof: Suppose (x6 is not an 8-1 automorphism. By 5.15,

there exists divisible D1, l.< Dl < D, such that

= D o

-l
B 03
D1 1

Thus

5 )

l = rank D < rank D. Thus by 5.15, a is

and clearly rank (D 1

not an 8-1 automorphism.
The.following result is due to Cernikov [11; Thm. 5.2].

Lemma 5.15: Let A be a finite group of 8-1 automorphisms of
D. Let q be the smallest prime divisor of [A] (q not necessarily

distinct from p). Then r < q.

Proof: Without loss of generality, r 2.2. Write D = P1 + P2,

P1 a group of type p3° and P2 # 1. Let a e A, la] = q. Now

<Ia >
Pl - D
< a >
and hence rank(Pl ) = r. Let Pl have generators al, a2, ...
and defining relations pal = O, pan+1 = an. Let 3n =

an + a: + -°° + dz . If only finitely many of the sn are trivial,

< snln = 1, 2, ... > is an a-invariant subgroup of D of type pm.

:8

Thus infinitely many of the sn are zero; but Since psn+1 n

we have Sn = O for all n. Hence

1 J
P: _<_<P(:]O_<_J'Eq-1;J°7£i>o

Thus r E'q - 1, as desired.

5%

We obtain a complete description of S-I automorphisms of order

p which depends on the following two lemmas.

Lemma 5.16: Let r < p - 1. Then D has no nontrivial auto-

morphisms of order a power of p.

Proof: Suppose otherwise. Then D has an automorphism. a of

order p. (1 satisfies
P _
x - l _ (x-l)mp(x) .

Viewing a as an r X r matrix over FP, <1 has minimal polynomial,

f, over F and
P
f|(x-l)ep(x) .
Since X - 1 and ¢p(X) are irreducible over Fp,
f e {X-l, spec), (X-l)cpp(x)} ;

i.e., deg f e [1, p-l, P). Let g be the characteristic polynomial

of’ a.
des s = r < P - l ,

and we have flg since g e FP[X]. Thus deg f = 1; i.e., f = X - 1

and, a, is trivial, a contradiction.

Corollary 5.17: Let P be a Cernikov p-group. If D, the
minimal subgroup of finite index, has rank r < p - 1, then D _<_ Z(P);
in particular P e ‘7 . (This result is attributed to Cernikov by

Blackburn [#3 Introduction to Sec. 5], but his reference is spurious.)

55

Proof: D E Z(P) follows from 5.16. Then P/Z(P) is an image

of the finite p-group P/D and hence is nilpotent. Thus P e 8'! .

Theorem 5.18: Let 0: be an automorphism of D of order p.

Then a is an 8-1 automorphism if and only if r = p - 1.

Proof: Necessity. By 5.15, r S p - 1. If p = 2, we are
done. Assume p > 2. Suppose for contradiction that r < p - 1.
Then by 5.17 D]< a > e 11 and by 11.8, 2 = Z(D]< a >) is infinite.

Hence Z n D is infinite and clearly
<Ia >

Since a is an 8-1 automorphism, Z n D is not proper in D, so

D < Z. But this means (1 acts trivially on D, contradicting lozl = p.

Thus r = p - l, as desired.

Sufficiency. Let Oi be an automorphism of order p and let

 

r = p - 1. Choose D minimal such that D is a divisible subgroup

1 l
of D and DC: = Dl° By the minimality of D1 and by 5.15, o is
an 8-1 automorphism of D1. By the necessity just proved, rank D1 =
p-l=rankD=r. Thus DlzD.

It is easy to verify that if a e GL(p-l, RP) has rational

canonical form

(*) o = .

 

 

)(P-1)X(P-l)

56

then (1p is the identity matrix. By the theorem just proved we have
that (am is an S-I automorphism of D in the case r = p - l, where

Q is the mapping of 5.9 and 5.10. The converse is also true:

Theorem 5.19: Let <1 6 GL(p-l, RP), with. up an S-I auto-

morphism of D of order p. Then. a has rational canonical form (*).

2322:: Since T is an isomorphism, (1 has order p and thus
satisfies the polynomial Xp - l = (X-l)¢p(X). Since a does not
satisfy x - 1, a does satisfy op(x). By 5.7(iii), mp is irre-
ducible over Fp so that ¢P is the minimal polynomial for (1.

Since the degree of ¢p is p - l, ¢p is also the characteristic

polynomial for a; the desired form (*) now follows.

Remark 5.20: Using essentially the same proof as for 5.7(iii)

it can be shown that the polynomial

2

p
o2(x)=2(_§_:_l
P X-1

is irreducible over FP. Thus if (1 is an automorphism of D of
order p2, then <xp is not an 8-1 automorphism of D. For ¢ 201) = O

P

and by the irreducibility of G 2 we have ¢ 2 divides the character-

P P
istic polynomial, f, of (1. But then

rankD=dengdeg¢2=p(p-l)>p-1
P

and by 5.18, a is not an 8-1 automorphism. .A consequence of this
observation is that for P, D, and C as in 5.1h or h.h we have

IP:C] = p. We have recorded this fact for the situation of 5.1h in

6.2.

Chapter VI
Relationship to Direct Limits of p-Groups

of Maximal Class

There are precisely two nontrivial direct limits of p-groups
of maximal class and presentations for them are known. Both are

Cernikov p-groups G, satisfying
[G:zw(G)] = [Zn+1(G)‘Zn(G)] = p

and the rank of D(G) is p - 1. (See [h; Sec. 5] and [5].) In

this chapter we use some characterizations of Blackburn for such groups
to show a relationship with the groups we studied in Chapter III;

then we use the results of Chapter V to obtain another characterization
of direct limits of p-groups of maximal class. Finally we provide

some examples using the external semi-direct product with an amalgamated

subgroup.

Theorem 6.1: [h; Thm. 5.1]. Let P be a Cernikov p-group
for which D(P) has rank r f_p - 1. Then either G/Z(G) is finite
or G has a finite normal subgroup N such that G/N is a direct

limit of p-groups of maximal class.

Theorem 6.2: Let P be an infinite p-group. Then
P 6 L81 (1 Iflsnb - )1 if and only if there exists a finite normal
subgroup N”< P and a normal divisible abelian p-group D of rank
jp - 1 such that P/N is a direct limit of p-groups of maximal class

and CP(D) = DN. Furthermore, in this case, lean = p.

57

58

Proof: Necessity. By 5.1h, D = D(P) has finite index in P
and has finite rank and P/C (where C = CP(D)) is isomorphic to a
nontrivial p-group of S-I automorphisms of D. By 5.18 rank D = p - l.

snb - I! so that by 6.1 there

Now Z(P) is finite since P e If!
exists a finite normal subgroup N <1 P such that P/N is a direct

limit of p-groups of maximal class. By 2.15 [D,N] = 1; i.e.,

DN 5 C. Now DN/N is a normal subgroup of P/N and since

2!. D
N Din N

"2

 

DN/N is a divisible abelian p-group of rank p - 1. Thus

and hence IP:DN| = p. But 0 < P so that
p E.IP‘C' f IP:DNI = p

and we have DN = C.
Sufficiency. Let N and D be as in the statement and let
C = CP(D) = DN. P/N is Cernikov and N is finite; hence P is a

Cernikov p-group. Now

 

shows that C/N is a normal divisible abelian subgroup of P/N of

rank p - 1. Thus

and we have

P DN 13.9-2.2-
[fi' TV] - lfi’o N] — lN" Zqu)] — P .

Notice that IP:DI = IP:DNI IDN:D| < co so that D = D(P). By a well
known theorem of Fitting, WC = DN e If, and hence condition 5.1h(i)
is satisfied. But P/C is naturally isomorphic to a subgroup of
Aut D and has order p. Thus by 5.18, P/C is a group of 8-1 auto-

morphisms of D and thus condition 5.lh(ii) is also satisfied. Hence

P eLfln Insnb- TI.

Now we obtain a characterization of direct limits of p-groups
of maximal class which leads to the construction of examples of the

groups studied in Chapters III and IV.

Theorem 6.5: Let G be a p-group, p 2_5. The following are

equivalent:
(i) G is a direct limit of p-groups of maximal class.

(ii) G is a semi-direct product of a divisible abelian group
D of rank p - 1 by a cyclic group of order p or p2
with an amalgamated subgroup of order 1 or p, respec-

tively, and is nonabelian.

Proof: (i) implies (ii): Let G be a direct limit of p-groups
of maximal class. Then [G:Zw(G)[ = IZn+l(G):Zn(G)] = p, n = 0,1,2,... .
jFurthermore, ZaHG) is a direct sum of p - 1 groups of type p”.
ILet D = ZufiG). Let x e G - D. Since IG:D] = p, we have
G=D<x>. Now lG:Dl=ID<x>:Dl=l<x>:Dn<x>|=p. Thus

D n < x > = < xp >. But now notice that

<xP>=Dn<x>ch(D)n CG(x) =Zl(G) .

60

Thus I< xP >] divides IZl(G)] = p; i.e., I< xP >] = [Dn< x >]
islw p H'Bn<x>=<fl>=1,tMnlfl=p,ifbn<x>

p> has order p, then [x] =p2.

= < X
(ii) implies (i): Let G be as in (ii). Let < x > be of
order p or p2, with Dn< x > trivial or of order p, respec-

tively. Then D <1 G and
IG:DI = [D<x>:DI = I<x>:D/l<x>[ = p .
The proof is complete by a theorem of Blackburn:

Theorem 6.h: [h; Lemma 5.2]. If G is a nonabelian p-group
with a normal subgroup D of index p, and D is the direct sum
of at most p - 1 groups of type p”, then G is a direct limit

of p-groups of maximal class.

In order to construct examples for 6.5 (and hence also for 6.2)

we make use of the following construction.

Lemma 6.5: Let M, D, and K be groups, M _<_ D and let
W:K -> Aut D be a homomorphism and 9:M -> K an isomorphism into K,
with M invariant under K1]! and satisfying

(i) d(m6)v = dm for all (1 e D, m e M, and

(ii) (mw)6 = (m9)k for all m e M, k e K.

Then there exists a group G which is a semi-direct product of D
by K with an amalgamated subgroup M such that D, K, and M are
isomorphic respectively to D, K, and H, and the action of K on D

corresponds to the action of KW on D.

61

Proof: Let G be the semi-direct product of D by K with

l e Glm e M}. We argue exactly as in

respect to v. Let N: {mem-
Gorenstein [21+; pp. 27-28] that N is a normal subgroup of C. Set

G: G/N and D, K, and M the images of D, K, and M, respectively.
Gorenstein's arguments show that D g D,

H
171 E D n K. He then concludes that H = D n K from the finiteness of

. 'A routine argument works

 

the groups by noting that [M] = ID I] K

for the infinite groups considered here:

Let xNeflllrlflK=-12-l\-II\I 3%. Write xN=dN=kN, for some
1

d e D, and k e K. Then k'ld e N. Let m e M, such that k- d

(m9)m-l. By the uniqueness of representation in KD, we have

k =m6, d=m eM. Hence xN=dNeMN/N=M. Thuswehave

UI

D “HEM as well so that H = Dr] K. That the action of K on

corresponds to the action of KW on D follows easily.

Example 6.6: Let D be a divisible abelian p-group of rank
r = p - l, and 01 an automorphism of D of order p. Then

D]< a > is a direct limit of p-groups of maximal class by 6.11.

C]
II

Example 6.7: Let D D + D + --° + D be a divisible

l 2 p1

abelian p-group of rank r p - l, with Di generated by xi n’
)

n = 1, 2, ... satisfying the relations pxi 1 = 0, px
)

. =x.
l,n+1 1,n’

n = 1, 2, ... . Let on e GL(p-l, RP) be the matrix

0 l
0

 

 

.)(P-1)X(P-l)

62

View a as an element of Aut D, with respect to the decomposition
above. Note la] = p. Let K: < y > be cyclic of order p2 and
v:K » <la > the homomorphism for which yt = a. Notice ypw is the

trivial automorphism. Now let

M= ((p-l) (p—e) 2 1) 2’1 < D .

Notice that M is cyclic of order p. We claim that M is invariant _

under a:

((P-1) (P-2) --- 2 1) 2’1 a

 

 

(-1 (p-e) (p-s) ... 2 1) 2’1

xp-l, l

x1,1

((P-1) (P-2) (P-5) --~ 2 l) i

ll

xp-l, 1

Thus a in fact acts trivially on M. Now let 9:M —: < y > be the

homomorphism defined by

((P-1) (P-2) -°- 2 1) t e = yp .

 

Xp-l,l j

Notice that new is trivial so that condition (i) of 6.5 is satisfied.
Since G acts trivially on M, KW does also, and since K is abelian,
condition (ii) of 6.5 is also satisfied. Thus the semi-direct product

G of D by K with M amalgamated exists; by 6.5, G is a direct

limit of p-groups of maximal class.

Example 6.8: Let G be the group of 6.6 or 6.7. Let N be

any finite p-group. Set P = G + N. By 6.2, P e L” e Ifisnb - )7.

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