ON. A GEN£RALIZATION or
SUBNORMALITY IN INFINITE GROUPS

Thesis for the Degree of PII. D.
MICHIGAN STATE UNIVERSITY
ANTHONY IOHN VAN WERKHOOVEN

1973* V

 

Michigan State
Univemt)’

h.

   
     

This is to certify that the
thesis entitled
ON A GENERALIZATION OF
SUBNORMALITY IN INFINITE GROUPS

presented by

Anthony John van Werkhooven

has been accepted towards fulfillment
of the requirements for

_Ph.-_D-_ degree in Mathematic s

iffW

Major professor

Date April 6, 1973

 

07639

 

ABSTRACT

ON A GENERALIZATION OF
SUBNORMALITY IN INFINITE GROUPS
BY

Anthony John van Werkhooven

The concept of an f—subnormal subgroup is defined
by R. E. Phillips in [1]. We say that the subgroup H is
an f-subnormal subgroup of G (written H<1f<1G) if there

exists a series

S:H=HOCH1;...;Hn=G
of finite length such that either Hi<1Hi+l or |Hi+lei|<:a.
We say that the series S is an f—series for H in G .. In

this paper a number of questions concerning f—subnormal sub-

groups are investigated.

We study conditions under which the join of two
f—subnormal subgroups is an f—subnormal subgroup. It is
shown, for example, that whenever G is metabelian, finite-
by-abelian, nilpotent-by-abelian, or FC—by—abelian, then the
join of two finite f—subnormal subgroups of G is a finite
f-subnormal subgroup. H. Wielandt showed that the join of
two finite subnormal subgroups is a subnormal subgroup. In

contrast to this result, we exhibit an abelian-by-finite

Anthony John van Werkhooven

group Which has two finite f-subnormal subgroups whose join

is not an f—subnormal subgroup.

In the main result of Chapter IV we show that under
certain restrictive conditions on the class 1, the join of
finitely many f-subnormal z-—subgroups is a z-subgroup.

This result implies that the join of finitely many f-sub-
normal solvable-by-finite-, 911-, {ISIS—subgroups is a solvable—

b
V A V A
by-finite-, mI-, m%-subgroup respectivelyfi, where mg(m%)

s
is the minimal (maximal) condition for subnormal subgroups.
V

In Chapter V f—subnormalSRS-subgroups are studied.
It is shown that if H and K are f—subnormalENs-subgroups
of the group G, then |HI:NH(K)| < m . This result is
used to prove that if 1 is a class closed under the taking
of subgroups and homomorphic images, then the following are

equivalent:

(1) If Gezz, the join of two finite f—subnormal
subgroups of G is a finite f—subnormal subgroup.
V
(2) If G e I , the join of two f—subnormal {ms - sub—
V
groups of G is an f-subnormal sms - subgroup.
A generalization of the concept of a nilgroup is

investigated briefly in Chapter VI. We say that G is a

Bf-group if for all xeG, <x><If<IG . A description of

Anthony John van Werkhooven

Bf-groups in which the length of the shortest f—series is
bounded by l or 2 for each element is given. It is

V
shown that Bf-groups which satisfy IRS are periodic groups

in which 1G :Z(G)\ < w .

1. Phillips, Richard E., "Some generalizations of normal

series in infinite groups." To appear in the J. Austral.

 

Math. Soc.

 

ON A GENERALIZATION OF
SUBNORMALITY IN INFINITE GROUPS

BY

Anthony John van Werkhooven

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Mathematics

1973

To

Linda

ii

 

ACKNOWLEDGMENTS

I would like to eXpress my sincere thanks to
Professor Richard E. Phillips and Professor Lee M. Sonneborn
for their encouragement and many helpful suggestions during
the preparation of this manuscript. I am especially grateful
to Professor Sonneborn for his interest in my progress during

my stay at Michigan State University.

iii

TABLE OF CONTENTS

Page
Chapter I. Introduction and Notation . . . . . . . . . 1
Chapter II. f—Subnormal Subgroups . . . . . . . . . . 8
Chapter III. The Join of f—Subnormal Subgroups . . . . 20
Chapter IV. The Join of f-Subnormal 13-Subgroups . . . 31
Chapter V. f—SubnormaleS-Subgroups . . . . . . . . . 38
Chapter VI. Bf—Groups . . . . . . . . . . . . . . . . 48
Bibliography . . . . . . . . . . . . . . . . . . . . . 55

iv

Chapter I

Introduction and Notation

In this paper we will investigate a generalization
of the concept of a subnormal subgroup. The definition is

due to R. E. Phillips, [7].

Definition 1.1: We say that .H isan f—subnormal

 

subgroup of G (H«df‘aG) if there exists a finite chain of

subgroups
H = HO ; H1 g .. . ; Hn = G
such that either Hi<:IHi+ or \Hi+l:.Hi| < o . We refer

1

to such a series as an f—series for H in G

H. Wielandt defined the concept of a subnormal

subgroup in [17].

Definition 1.2: We say that H is a subnormal

 

subgroup of G (H<:<CH if there exists a finite chain of

subgroups

H=HO<IH1<I Hn_14Hn=G

Notation 1.3: If H is a subgroup of G, we write

H c G and denote by |GI:H| the index of H in G . If

H is a normal subgroup of G, we write H‘OG . If N is
a subset of G, NG denotes the normal closure of N in
G . When N = {x}, we write xG for {X}G If H c G,

then H9 denotes the conjugate of H by gng and

Core (H) = (I H9
G
96G

If H and K are subgroups of G, then CK(H)
denotes the centralizer of H in K and NK(H) denotes

the normalizer of H in K .

If HeddCS, it can be shown that there exists a
canonical normal series from H to G (see for example
[8] or [10]). This canonical normal series from H to
G is referred to as the standard series for H in G . It
is well known that the standard series for H in G yields

a normal series of shortest length from H to G .

Definition 1.4: If H<<G, the length of the

 

standard series for H in G is called the index of

subnormality for H in G and is denoted by s(G,H)

H. Wielandt proved in [17] that whenever G has a
composition series, the join of two subnormal subgroups of

G is again subnormal in G . In contrast to this result,

D. Robinson has given in [10] examples of groups in which
the join of two subnormal subgroups need not be a subnormal
subgroup. Various conditions under which the join of two
subnormal subgroups is a subnormal subgroup are discussed,
for example, in [l], [8], [9], [10], [ll], [12], [14], and

[17].

In this paper we establish a number of sufficient
conditions under which the join of two finite f-subnormal
subgroups is an f—subnormal subgroup. In Theorem 3.18 we
give an example of a group G which has two finite
f-subnormal subgroups whose join is not an f-subnormal
subgroup of G . One may compare this result with Theorem 7
of [17], in which H. Wielandt shows that the join of two

finite subnormal subgroups is always a subnormal subgroup.

Notation 1.5: The unit element and the group of

order one will both be denoted by 1 .

Definition 1.6: A (group theoretic) class F is a

 

collection of groups such that l eT‘ and whenever Gear

and G1 is isomorphic to G, then Gle:r

Notation 1.7: (i) We will use the following nota-
tion for those classes of groups which will be referred to

frequently.

 

(ii) If r1

abelian groups

- solvable groups with derived length less

than or equal to n

solvable groups

finite groups

nilpotent groups (of class s c)

finitely generated groups

groups satisfying the maximal condition
for subgroups, normal subgroups, subnormal
subgroups

groups satisfying the minimal condition
for subgroups, normal subgroups, subnormal

subgroups

and r2 are two classes of groups,

than rlrz is the class defined by

rlt‘z = [G IaNaG, Nerl, and G/NeI‘Z]

(iii) If F

ST

QT

is a class of groups, then

[H|C~1GeI‘ and HcG]

{HISG€T, N<IG, and G/N = H]

sn 1* = [H‘BGeI‘, H<I<IG, and [6:11] < a}

0

(iv) If F1

G€rlvr2

and r2 are two classes of groups, then

if GeI‘l and Gefz:

if Gel“l or GeI‘2

 

We define the property (*) in Definition 4.4. We
say that the class 1 has the property (*) if
(*) for any group G, the join of a finite number

of subnormal z—subgroups of G is a I—subgroup.

It is known for example that the class 6 has the

property (*) [l6 ; Theorem A].

We establish in Theorem 4.5 that if z = snoz is a
class of groups having the property (*), then the join of a
finite number of f-subnormal Ig-subgroups of a group G is

a Ig-subgroup of G .

In [11] and [12] J. E. Roseblade gives the following

definition.

Definition 1.8: Let I be any class of groups. We

 

say that z is a subnormal coalition class if, given that
H and K are subnormal x-subgroups of G, the join <H,K>

is a subnormal z-subgroup of G .

It is clear that if z is a subnormal coalition class,
then 1 has the property (*)
V
In [11] Roseblade showed that m% is a subnormal
coalition class and in [12] he showed that $5 is a subnormal

coalition class. Further results on subnormal coalition clas—

ses may be found in [8], [9], and [14].

v
In Chapter V we study f—subnormal EmS-subgroups of

a group. We establish in Theorem 5.8 that if H and K are

f—subnormal gg-subgroups of G, then [H :NH(K)] < m ,

where NH(K) denotes the normalizer of K in H . We a
showed in Corollary 4.12 that the join of finitely many

f—subnormal 33/18 - subgroups is a 5‘s - subgroup. We use

Theorem 5.8 to give an alternate proof of Corollary 4.12.

In Theorem 5.14 we establish that for certain classes
of groups, the join of two finite f-subnormal subgroups is
always an f—subnormal subgroup if and only if the join of two
f—subnormal %S-subgroups is always an f—subnormal subgroup.

R. Baer defined in [l] the concept of a nilgroup.

We generalize this concept in Definition 6.2. We say that
G is a ‘Bf-group if for all xeG, <x> <If<G . In
Propositions 6.4 and 6.6 we examine fif-groups for which
there is a uniform bound on the length of the f-series for
each element. In Proposition 6.15 we use the results of
Chapter V to characterize Tf-groups with ENS

Notation 1.9: If N and M are subsets of G ,

we denote by (DL Nj> the subgroup of G generated by N

and M . We denote by [M,N] the subgroup

<:[m,n] [mg;M, right),

where [m,n] is the commutator of m and n

f
Notation 1.10: We will often indicate by H c K

that H is a subgroup of finite index in K .

Notation 1.11: If G is a split extension of H

by K, we write G = H]K .

Notation 1.12: We will denote by Cpub the group
defined by the set of generators X = {xl,x2,. ..] and the

. _ p _ p _ .
set of relations R — [xl — 1, xi+l — xi] 1 s 1} .

Chapter II

f—Subnormal Subgroups

It is inherent in Definition 1.1 that an f—series
for an f-subnormal subgroup H of a group G is of finite
length. Hence, for every f-subnormal subgroup H of G,
we can find an f-series of shortest length. We denote the

length of such an f—series by f(G,H).

Remark 2.1: If H<If<IG, then f(G,H) is an
invariant of the pair (G,H). This seems to be the only

meaningful invariant of (G,H).
Suppose that f(G,H) = n and

S:H=HOCH1C. ..CHn=G

is an f—series for H in G . We can define functions 91
and 92 which have as their domain all triples (G,H,S),
where H‘df‘QG and S is an f—series for H in G of

length f(G,H). Let

the number of non-normal "jumps” in S

91(G,H,S)

and

92 (G,H,S) : ::{‘Hi+l :Hi‘ ‘Hi flngf-l and ‘Hi+l :Hl‘ 'i a: }.

I..Y_"."'.._-_ . _—r'

If I.

 

Example 2.2 shows that neither 91 nor 92 is an invariant

of (G,H)

Example 2.2: Let

N z:

1 <(x(l,i)‘> and N =

1 2 i

<X(2,i)\,
1

IIMB

HMS

i
where the elements x(i,j) are of order 2. Let N be the

direct product of N1 and N2 ,
N=N1®N2

We define the automorphisms t1 and t2 of N by

t1 t1
x(l,i) = x(l,i) x(2,i) = x(l,iy+x(2,i)
t2 t2
x(l,i) = x(l,i)-+x(2,i) x(2,i) = x(2,i)
Let G = N ]< tl,t2> . It is readily verified that t1 and
t2 are automorphisms of order 2 and that
< tlst2> 3‘ S3 1
the symmetric group on three letters. Consider the f-series

Sl:<X(l,1)»<IN<IG

and

2 °° . f
S :<X(1,1)><I(<XI1,1)>e 23 <X(1,1)>)®N2:G
i=3

for <jx(l,1)j> in G . Since < x(1,1)j> is neither normal

10

 

nor of finite index in G, f(G, <x(l,1) >) 2 2 . The
existence of the f—series S1 implies that f(G,< x(1,1) >)= 2 .
Hence S1 and 82 are f-series of length f(G,<x(1,1)>) .

We note that

ll
H
L

l
gl(G,H,S I o gIIG,H,sz)

l
gZIG,H,s > = o 92(G,H,SZI 12

It follows that neither 91 nor 92 is an invariant of the

pair (G,H).

The following two lemmas are easy consequences of

Definition 1.1.

Lemma 2.3: If H<If<G and KQG, then Hande

and f (K, mm) s f (G, H).

Proof: If f(G,H) = n, then there exists an f—series
H = HOcch. . . CHn = G
for H in G . Let us consider the series
(a) HnK = BOcBchzc. . . cBn = K ,
where B. =H.flK for Osisn . If H. <IH. , then
1 1 1 1+1
Hinxaniflnx. If ‘Hi+1:Hi] < co, then
”11+an :HinK| s ‘Hi+l : Hi] < a: . Thus the series (a) is an

f-series for HnK in K. Hence HrIK<If<IK and f(K,HnK)sn CI

11

Lemma 2.4: If H<If<IG and K<IG, then HK/K<f<IG/K

and f(G/K, HK/K) sf(G,H)

Proof: If f(G,H) = n, then there exists an f—series
H=I%CH1C.. Cm1=G
for H in G . Let us consider the series

(a) HK/KcHlK/‘Kc. . . :HnK/K = G/K .

If Hi 4Hi+l , then HiK/K<Hi+1K/K . If [Hi-+1 : Hi] < co ,
then ‘Hi+lK/K : HiK/K| s |Hi+1 : Hi] < co . Hence (a) is an
f—series for HK/K in G/K . We conclude that HK/K<Jf<JG/K

and f(G/K, HK/Kk;f(G,H) = n . E

In what follows, frequent use will be made of Lemmas

2.3 and 2.4 without further reference.

Remark 2.5: A subgroup of finite index in a subnormal
subgroup of G is an f—subnormal subgroup of G . Also, a
subnormal subgroup of a subgroup which has finite index in G
is an f-subnormal subgroup of G . we will show in Example
2.8 that not every f-subnormal subgroup of G need be of

either of the above types.

We will need the notion of a wreath product in

Lemma 2.7.

 

12

Definition 2.6: Let H and K be groups and let

 

B = z Hy; be the direct sum of isomorphic copies of H
keK
indexed by the elements of K . Let bqu with b(k) as

its k-th component. If xe;K, we define bX by the

1

rule bx(k) = b(kx- ) . The wreath product of H with K,

H wr K, is the split extension B:]K .
For further information see [6]

Lemma 2.7: If A is a simple group and B is

co

isomorphic to Cp , then A wr B has no proper subgroups

of finite index.

Proof: It will suffice to show that A wr B has no

proper normal subgroups of finite index. Suppose K«dA wr B

and ‘A wr B::K] < w . Since B is divisible and
\BI:BnK| < m , Bg;K . It will now suffice to show that for
B
some bOeB, AbOCK, for then (Abo,B>=AercK and
K = A wr B . Since 2 Ab is an infinite group and
beB
I 2A,,:( 2A )flK\<oo,
1333 1333 b
there exists 171x e< ZAb)flK. Let b 6B such that
O
bgB
x(b ) i'O . Let yI;A such that [)(U)), y] # l . Using
0 b0 0 .

the fact that

EAbjflKq

X A ,
b63 6 b

b B

 

 

13

we observe that

.1y
1% x(b),y =X X a 2A K
I O I (mum

and

17(x-lxyeAb flK .

0

Since A []K<3A and A is simple, it follows that

b b b

O O O
A CK. Hence K=Aer. E]
bO

Example 2.8: Let Ai be the alternating group on 1
letters and let C be an arbitrary infinite simple group.

Let G = C'XSA5 be the split extension

IIMU'I

Ii

1C1) 1A5 ’

where Ci=-C and the elements of]% act as permutations on

S .
the coordinates of the elements of 2 Ci . Let
i=1
4 G A5
H = c1425.4 = [ iglci /‘]A4CG . Slnce H 2A4 =A5,
G A5 '
II 2<1C1 ,A5j> = G . Hence, H is not a subnormal subgroup
of G . We note that H is an f-subnormal subgroup of G
since
5

._ . f
(a) H — C'\,4A4<IK iglci) A4<;G

is an f-series for H in G . Since lG :H| is not finite,
f(G,H) > 1 . It follows that f(G,H) = 2, since the series

(a) shows that f(G,H) 52

 

Let G: Ger, where Ba-Cpoo and p is a prime

 

 

14

number. If we identify G with G we observe that

1’

H<af<aG
since

H<3f<3G
and

G<1 Z Gb<aG
beB

Since H is not a subnormal subgroup of G, H is not a

subnormal subgroup of 'G .

We will now show that 5' contains no proper subgroups
K of finite index in 5'. If we denote by A5(b) the sub-
group of Gb corresponding to the subgroup A5 of G, then

the subgroup [ z A5(b) ) -B of G is isomorphic to
bgB

A5 wr B . It follows from Lemma 2.7 that [ bZB AS (b)/

has no subgroups of finite index. Hence K er A 5(b))-IBC;K .

If we denote by Cl(b) the subgroup of G bcorresponding to

b
the subgroup C1 of G, then the subgroup K 23 C1(b))- B
1353

of G is isomorphic to C1 wr B . Again, it follows from
Lemma 2.7 that K 2. C 1.(b))°B c K . It now follows that
bgB
< 213503), FC1(b)>= 2(3ch
bgB beB beB
and
G = Gin) 1B = K .

beB

 

 

 

15

we conclude that GI contains no subgroup K such that

H<IQK and |G:K\ < co .

Suppose that L is a subgroup of G. such that

L<I<G and ‘L:H|<ao. Then LnGl=LflG<I<IG and
f
|LflG :H] < on . Hence H r; LflG<I<I G . We showed that
HG = G . It follows that (IJWG)G = G and LFWG = G . Since

]G»:H] is not finite, we conclude that the subgroup L can-

not exist.

It will be of interest to know when an f—subnormal

subgroup is a subnormal subgroup.

Definition 2.9: Let r be a class of groups. We

 

say that G belongs to the class LP if every finitely

generated subgroup of G is contained in a F-subgroup of (3.

Lemma 2.10: If G is an Lfn— group, then H<af<3G

if and only if H<I<JG .

Proof: If H<If<IG, let

Ii==I%)c chz. .. C Hn = G
be an f—series for H in G . Suppose that
‘Hi+1‘Hi‘< °° ’

then

L = Hi+1/CoreH. (Hi)
1+1

 

 

16

is a finite Lm— group since the class Lin is closed under
homomorphic images [4;p. 222] . Hence Legm . It is well

known that in a nilpotent group every subgroup is subnormal

[4:p. 225] . Thus
H./Core (H.)<a< H. /Core (H.)
1 Hi +1 1 1+1 Hi+1 1
But then Hi<I<IHi+l and H<I<IG . D

The following lemma shows that in a certain sense
f—subnormal subgroups are not too far removed from being

subnormal subgroups.

Lemma 2.11: If H<If<)G, then there exists quG

such that H c H, ]H:H] < co , and s(G,H) s f(G,H) .

Proof: We will prove the lemma by induction on

f(G,H).

Suppose f(G,H) = l . If H<aG, there is nothing to

prove. If [G :H] < m, we may choose H'= CoreG(H) .

Let us now assume that the assertion is true for

n-l and that f(G,H) = n . Let

H = Hocch. . .c:Hn = G

be an f-series for H in G . If Hn_1<aG, then

f(H H) = n—1 and the induction hypothesis yields the

n-l’

 

r;

17

existence of fiddHn-l such that H DH, [H :H] < co, and

s(H H)sn—l . But then quG and s(G,H)sn

n—l’

If ]G :Hn-l‘ < m, we let L = CoreG(Hn_l) . Now

let us consider the series
(a) HflLeHlnLe. . . cHn_1flL = L .

then HifgLGHi+lflL and if [H -Hi|<ao,

If HiQHi-tl’ 1+1 -
lHi+1r]L:lHi()L| < m . Thus, (a) is an f-series for HFWL
in L . Since f(L,H[]L) < n, the induction hypothesis yields

the existence of H44 L with the property that [HflL :HI < a:
and s(L,H) sn-l . Now, ]H:HflL| < a: and L4G imply the

desired conclusion. C]
A useful consequence of Lemma 2.11 is the following

Corollary 2.12: If H is an infinite f—subnormal

 

subgroup of G which has no proper subgroups of finite index,

then H<I<IG. [3

Definition 2.13: If f is a class of groups, we

 

say that 2 is a radical class if 52 = Z and every group
G has a normal $-subgroup, 2(G), containing all the normal
Z-—subgroups of G . We refer to f(G) as the V—radical

of G .

Lemma 2.14 [7; Theorem 3.1]: If 2 is a radical

class such that z = 23 and H is an f-subnormal subgroup

 

18

of G, then f(H) is an f-subnormal subgroup of 2(G) . D

It readily follows from Definition 2.13 that
Uig)g = LE5 is a radical class. Hence, we have as a conse-

quence of Lemma 2.14

Lemma 2.15 [7; Corollary 3.1]: If H and K are
locally finite f—subnormal subgroups of G, then <1L1C>
is locally finite. In particular, the join of finitely many

finite f-subnormal subgroups is finite. D

Next we present a few comments concerning the inter—

section of f-subnormal subgroups.

Proposition 2.16: If H and K are f-subnormal

 

subgroups of G, then HflK<f<IG and

f(G,HflK)s f(G,H)+f(G,K). [:1

Proposition 2.16 follows readily from Lemma 2.3.

Corollary 2.17: The intersection of a finite number

 

of f—subnormal subgroups of G is an f—subnormal subgroup

of G . D

The intersection of an arbitrary set of f—subnormal

subgroups of G need not be an f-subnormal subgroup of G .

This is shown in the following example.

.Imw

 

 

19

Example 2.18: Let G be the infinite dihedral

group
G = <a,b|bab = a—1,b2 = 1)
2i
Let Hi = (a ,b>, then Hi<l<lG,S(G,Hi) = i, and
nHi=<b>
i=1

Suppose (13> is an f—subnormal subgroup of G,

then there exists an f—series

<b> =AOCA1C...CAn==G.

Since Al 3 A

+. 0’

m i m . .

a b egAl . Hence a 9A1 . Let nO be the least pOSitive
. n0 n0 n0
integer such that a 6A1 Then A1 = <23 ,b|ba b==a
Since A1 is an infinite group, <t>><oA1 . But this cannot

no zno
occur since ba = ba 5 <13) . Hence <13> is not an

f—subnormal subgroup of G .

there is an element ambl,rn§¥0 such that

 

 

 

Chapter III

The Join of f-Subnormal Subgroups

It is well known that the join of two subnormal ' I

subgroups need not be a subnormal subgroup. It can be shown,

 

using the example given by D. Robinson in [8], that there
exists an mT2-group in which the join of two subnormal
subgroups need not be an f—subnormal subgroup. In this
chapter we study conditions which imply that the join of two

f—subnormal subgroups is an f-subnormal subgroup.

In the main result of this chapter, Theorem 3.18,
we give an example of a group in which the join of two
finite f-subnormal subgroups is not an f-subnormal subgroup.
One may wish to compare this result with Theorem 7 of [17]
in which H. Wielandt shows that the join of two finite sub-

normal subgroups of a group is always a subnormal subgroup.

The following lemma is an immediate consequence of

Lemma 2.4.

Lemma 3.1: If H<If<IG and K<IG, then

<H,K><If<IG and f(G,HK) s f(G,H). CI

Lemma 3.2: If H and K are f—subnormal subgroups

20

A‘I

 

21

of G such that K has an f-series

b) K=I%CK1C...Cfi1=G

and KiH = K. for i = 0,1,2, .. .n, then J = <ILIC><3f<3G

J.
a
and f(G,J):;f(G,H)° f(G,K) '

Proof: Let Ki and Ki+ be members of the f—series

1

(a) for K in G . If Ki<aKi+ then Ki<3HKi+1 . By

 

1’
Lemma 2.3, H<af<aHKi+l . An application of Lemma 3.1 shows i
that HKi<af<1HKi+1 and f(HKi+1,HKi):sf(G,H) . If
‘Ki+l: Ki] < m, then ‘HKi+l :HKil < m . Thus, in any case,
HKi<If<IHKi+1 and f(HKi+1,HKi) sf(G,H) . We conclude that
HK<If<IG and f(G,HK)sf(G,K)~f(G,H). [I

K
Lemma 3.3: If H<l<lG, dedG, and H. =H, then

<H,K‘><If<lG and f(G,<H,K>)ss(G,H)-f(G,K)

Proof: Let

H = anHn_1<I. . .<tHl<IHO = G
be the standard series for H in G . We will show by
induction on i that HiK = Hi . We note that HOK = GK = G .
s H K - H th H — HHi‘1 d h
uppose i-l — i-l’ en i — an ence
H-_ H._ H-_
HiK = (H 1 1)K = (HK) 1 1 = H 1 1 = Hi . An application of

Lemma 3.2 yields the desired conclusion. C]

22

Proposition 3.4: Let H and K be f—subnormal

 

subgroups of G such that HK = H and [K] < m, then

(mK><f<G aw fKL<mK>)sfmfifldHQK)

Proof: Let

(1).H=I%CH1CH2C...CHn=C3
be an f—series for H in G . Let
Bi = fl Hik
keK

Consider the series

(ii) H = BOQBchZ . . . GBn = G
If Hi<IH1+1, then H Hikq fl Hik+1 If 'Hi+1:Hi‘ < co,
keK keK
then I n Hik+l: fl Hik Is ‘Hi+l:Hi| 'Kl< co . Hence, (ii)
keK keK
is an f—series for H in G . Since BiK = Bi for 0:;is;n,

Lemma 3.2 implies that <H,K><If<IG and f(G,<H,K>) s

f(G,H)-f(G,K) . ‘ I]

As a consequence of Proposition 3.4 we have

Corollary 3.5: If G is any group and G1 is

torsion free, then the join of two finite f—subnormal subgroups

of G is a finite f—subnormal subgroup of G

Proof: Let G be a group such that G1 is torsion

free. If H and K are two finite f—subnormal subgroups

 

23

of G, then <H,K>e{5 by Lemma 2.15. Hence [H,K] = l
and HK = H . It now follows from Proposition 3.4 that

<H,K><If<IG. CI

Theorem 3.6: If H and K are two finite . I}

f—subnormal subgroups of G , then the following are equivalent:

(i) <H,K><If<IG

 

(ii) HK<If<IG

E“ I' .' ' L» \‘.'
.41

(iii) [H,K]<If<IG

Proof: Since H and K are finite f—subnormal sub-
groups of G, it follows from Lemma 2.15 that <H,K> 63

and consequently HK and [H,K] are finite subgroups of G .

Suppose that <H,K><If<IG . Since HK and [H,K]

are normal subgroups of <fLIC>, HK and [H,K] are
f-subnormal subgroups of G . Thus, (i) implies (ii) and (i)

implies (iii).

0n the other hand, if HK<1f<aG, it follows from
Proposition 3.4 that (HK,K> = <H,K§ is an f-subnormal

subgroup of G . Thus (ii) implies (i).

If [H,K]<af<3G, it follows from Proposition 3.4 that

< [H,K],H> = HquqG . Hence, (iii) implies (ii). [:1

Corollary 3.7: Let 1 be a class of groups with

24

the property

(A) If GIgz , then every finite subgroup of G is

f—subnormal in G

If quzm , then the join of two finite f—subnormal subgroups

of G is an f-subnormal subgroup of G .

2399:: If Gegzm , G has a normal subgroup N
such that Ne; and G/Nefl . Let H and K be two finite
f-subnormal subgroups of G . Since G/‘Nefll , [H,K] :N . It
follows from Lemma 2.15 that [H,K] €83 . Hence [H,K] <f<IN<IG
and [H,K]<1f<1G . It now follows from Theorem 3.6 that

(H’K>quGo U

Definition 3.8:

 

(i) We say that G is an FC-group (G€;FC) if for
every element x 3G, \G : CG(x)| < co .

(ii) For any group G we define the FC—center of G,
FC(G) , by

FC(G) = [x] ]G:CG(x)| < co }.

It is readily verified that FC(G) is a character-

istic subgroup of G .

Lemma 3.9: Every finite subgroup of an FC—group is

an f—subnormal subgroup.

 

'F—e

25

Proof: If G5;FC and H is a finite subgroup of
G , then \G :CG(h)| < m for every element h.;H . Since

H is finite,

G:C(H)> G:flC(h)snG:C(h) <0...
I G I ‘heHG lheHl G l

and

f
HcH-ch) s G

is an f-series for H in G . Hence, H is an f-subnormal

subgroup of G . C

Remark 3.10: The classes m,{b,m and FC satisfy
the property (A) of Corollary 3.7. It follows from Lemma 2.15
and Corollary 3.7 that whenever G 69101891, 3221, or (FCHJ, the

join of two finite f-subnormal subgroups of G is a finite

f-subnormal subgroup of G .

Proposition 3.11: Let H and K be f—subnormal
subgroups of G . If |G:NG(H)| < m, then <H,K><f<G
and f(G,<H,K>) 52 ° f(G,K)

Proof: Since [G :NC(H)| < m, [K :NK(H)| < m .

K

Let H = H . Then there exist elements k ,k ,. ..,k (3K
1 2 n
— kl k2 kn
such that H = (Ii , H ,. .. ,H >> . Since
n k. -
i n ‘—

|G : iQINGm )Is IG:NG(H)| < co , \G : NG(H)] < co . Hence,

—- —- f

H<IN (H) g G

G

 

 

26

is an f-series for H' in G . Since HK ='H and

NG(H)K = NG(H), Lemma 3.2 implies that

<H,K> = <H,K><If<IG
and '
f(G,<H,K>) 52- f(G,K) . [I I1

Corollary 3.12: If H and K are two f-subnormal

 

subgroups of the FC-group G and H 65, then <H,K> <If<IG . I! I

Proof: If H is a finite Subgroup of the FC—group
G, ]G :NG(H)] < a . Proposition 3.11 shows that

<H,K><If<IG. Cl

Definition 3.13: If we denote the center 2(6) of

 

G by 21(G), then for the ordinal number a we define

Za+1(G) by
ZO+I(G)/ZQ(G) = Z(G/Za(G)) .
If B is a limit ordinal, we define ZB(G) by
Z (G) = LIZ (G) .
6 0<6 0
We define the hypercenter of G,Zm(G), by

Zm(G) = LKZO(G)|<1 an ordinal number } .

We say that G is hypercentral if Zm(G) = G .

27

The following concept will be useful in the proof of

Proposition 3.15.

Definition 3.14: If

 

S:H=HOCH1C...CHn=G 1

is an f—series for H in G, we define n(G,H,S) to be the

number of factors in the series S which are not finite. “

Proposition 3.15: If G is a split extension of N

 

by H such that N511 and H58, then H<If<IG if and

only if Zm(G) = Zn(G) for some integer n and [G :Zm(G)] (:9.
Proof: Let

S:H=HOCH1C. ..CHm=G

be an f-series for H in G . If n(G,H,S) = k, we write

the f—series S

(i)H=H§H <1H CH <IH c...<_;H <IH CH =G,
B0 01 E31 02 32 0k Bk ak+l

where HBO/HO.(;},1sisk, and ma. :HB.|<m,Osisk
i 1 1+1 1

We proceed to prove the preposition by induction on
n(G,H,S). If n(G,H,S) = 0, G635 and the assertion is

trivially true.

We now state our induction hypothesis: If G is a

28

split extension of N by H such that N 621 and H63,
then for some integer n Zm(G) = Zn(G) and IE :Zm(G)| < m

whenever H, has an f-series 'S such that n(_;H;S) < k .

Now, suppose that G is a split extension of N by

H such that Neil, H38, and H has an f—series S as

given in (i) with n(G,H,S) k. Since Ho €25 and
1

II <11 | : c (H )| < e . Since [G :N] < m it
0‘1 51’ HE1 Hal 0’1 ’
follows that

Hi : ercH (Ha )] < m .

B1 61 1

But then \H :11 [12(6)] < m, since N{]C

(H ):Z(G)
Bl Bl 0

Hal 1
Let us now consider the group G/2(G) . G/Z(G) is a split
extension of NZ(G)/2(G) by HZ(G)/Z(G) such that

NZ(G)/Z(G) em and HZ(G)/Z(G) eg . The f—series

§:HZ(G)/Z(G) = H Z(G)/Z(G)t;HO Z(G)/Z(G)<)HBlz(G)/Z(G)...
1 -

Bo

H Z(G)/Z(G)<)H Z(G)/2(G)cH Z(G)/Z(G)=G/Z(G)
Gk Bk OLk+l

satisfies the inequality n(G/Z(G),HZ(G)/Z(G),§) < k . An
application of the induction hypothesis yields the existence
of an integer n such that Zw(G/Z(G)) = Zn(G/Z(G)) and
[G/Z(G) :Zm(G/Z(G))| < m . But then zm(G) = zn+1 (G) and

|G:Z°°(G)‘<°°o

29

The converse is a special case of Proposition 3.16.

Proposition 3.16: If G is a group such that

 

[G :Zn(G)I < a for some integer n, then every subgroup of

G is an f—subnormal subgroup.

Proof: Let H be a subgroup of G, then

H<I21(G)H<I... <Zn(G)H E G

is an f—series for H in G since IG :Zn(H)| < w and

zi (G)

H =<H,[H,Zi]>;<H,Zi_l>, lsisn. [3

Corollary 3.17: If G = N:]H such that N¢gfl and

 

H e {5, then H <) f <1 G implies that every subgroup K of G
is subnormal in a subgroup having finite index in G . Con-
sequently, every subgroup of G is an f—subnormal subgroup

of G .

Proof: If H<af<1G, then ‘G :Zk(G)| < c for some

integer k . For any subgroup K of G,

K<IK21(G)<I...<IKZ 2c

k
is an f-series for K in G . The corollary follows. E

(3)

Theorem 3.18: There exists a group G 521 A913
such that the join of two finite f—subnormal subgroups H

and K of G is not f—subnormal in G .

C]

 

30

ngpf: We let G be the group of Example 2.2. G is
a split extension of N = NlaN2 by < t1,t2>, where N 691
and <t1,t2:>=-S3, the symmetric group on three letters.
Hence, G¢;m(3)A9fi}. We recall that ti centralizes Ni and

N/Ni’ i = 1,2 . The subgroups < t1) and < t2). are

f—subnormal subgroups of G since
<t 4N t: N<t>fG
i> i< i><’ i 5
is an f—series for < ti> in G , i = 1,2

We now verify that Z(G) = 1. Suppose note;Z(G),

where n 5N and t €< t1,t2> . If t 7! 1, there exists
7t€<:t1,t2> such that tt ¢ t . But then (nt)t= nt and
n-lnt = t(tt)-1 . Since iNFh<tl,t2> = l, tt = t, contra—
dicting the fact that tt y t . Hence t = l . Since

nt = n' l = n 6N, nt= n1 +n2, where nl 9N1 an: n2 3N2 .
If nl # 0, then n 2 i n . If n2 #'0, then n l # n .
Hence n = 1 . We conclude that Z(G) = 1 . Now, Proposition

3.15 implies that <ftl,t2>. cannot be f-subnormal in G . D

Remark 3.19: If 1 is the class $(3),mg,(FC)g, or
n(FC) and Ge:z, then the join of two finite f—subnormal

subgroups of G need not be f-subnormal in G

Wm

41'

 

Chapter IV

The Join of f—Subnormal 1;}- Subgroups

In order to prove the main result of this section,

we will need the following technical lemma.

Lemma 4.1: Suppose HqG,]G:H| = n< a), and KGH.
If A = [a(l),a(2),...,a(n)] is a right transversal for H
in G such that 1 5A and G = <K,A> , then there exists

a finite subset L of H such that

a —1
H=<K,L(a €A> .

Proof: Let a,be A and keK . Since K:H<IG,
akb-lng if and only if a = b . For a(i), a(j) 5A,
l.si, j sn, we define a(i,j) 6A uniquely by the equation

a(i)a(j)a(i,j)_l¢;H .
Let H be defined by
H:(Ka,a(i)a(j)a(i,j)-l|a-15A,lsi,j_<.n\
Since H<IG,H§H. If geH, then

61 62 C

9 = 91 92 .. .gm

for some elements gi eKlJA and 6i = i-l,1.si.sm. Set a(io)=]..

31

32

There exists a unique element a(il) 5A such that

e c
. 1 . . 1 . -1 .
a(iO)g1 eHa(ll) . Hence a(lo)g1 a(il) EH .e It is
easily verified in fact that (a(io)gl'la(il)-l) 1 is a

(displayed) generator of H'. Suppose that for all j ,

l sj < 1, sm, we have chosen a (ij) 6 A such that ' l
C ' ._ ._ .1-
a(ij_1)gj Ja(ij) lng . We then choose a(i‘) 9A as the 1‘
6.
unique element satisfying the equation a(i‘_1)gl ‘5;Ha(i‘) .

6

. II . -1 e: ”‘7
Then (a(ll-l)g'! a(ll) )

is a (displayed) generator of
_ . C£ . -1 _
H and a(1£_1)g‘ a(il) 5H . But then

a: g: _
g = IaIiOIg1 1a(i1)_1)(a(il)gz 2aI12> 1). ..

(aIim_1Igm€maIim>“1>aIim>,

e _ _. ._
where a(i‘_l)gl taut) 15H, lszsm . Since HcH, it

follows that a(im) = l . Consequently, g'er and H = H .

The lemma now follows if we set
. . . . —l . .
L= [a(i)a(j)a(i,3) Ilsi,jsn]. [3

Definition 4.2: If x is a class of groups, we

 

define 6£(G) by
935(6) =<HIH<I<G and Hez>

We note that em(G) is the Baer radical of G

(see for example [8: p. 101]).

33

Remark 4.3: It is clear from Definition 4.2 that

for any class 1

(1) 91(6) is a characteristic subgroup of G and

(ii) whenever T is a finite subset of 9§(G), there
exist a finite number of subnormal z-subgroups H1,H2,. . .Hh an
of G such that

<T>§<HPHT"'”Hn> . "

Definition 4.4: We say that the class 1 has the

 

property (*) if

(*) for any group G, the join of a finite number of

subnormal z-sugroups of G is a z-subgroup of G .

We recall that, for an arbitrary class 1, the

class snoz is defined by
snoz =[H|H<1<)Gez and ‘G:H| < co} .

Theorem 4.5: If x = sn is a class of groups

02
which has the property (*), then the join of a finite number

of f—subnormal :3-subgroups of a group G is a zg-subgroup

of G .

Proof: Let H1,H2,...,H be f—subnormal zg-subgroups

n

of G . we may assume, without loss of generality, that

G = (H1,H2,. . "Hm-5 . Since Hi €35, there exists Ki <1Hi

 

34

such that ‘H.

1:Ki‘<°° and K161 for lsisn. We note

that Ki,l;si:sn, is an f—subnormal x-subgroup of G .
It follows from Lemma 2.11 that for each subgroup Ki there
exists a subnormal subgroup Fi of G such that Fic:Ki
and IKi:Fi|<oo. Since z=snoz,Fiez,lsisn. Since

|H.

1 : Fi‘< co and FiceI (G), it follows that HieI (G)/eI (G)

is a finite f—subnormal subgroup of G/GI(G) . An applica—

tion of Lemma 2.15 shows that
G/ez(G) = (H1, . ..,Hn>9z(G)/91(G)e‘{§ .

Let T and Ti be right transversals for 91(6) in G and

Fi in Hi’l si sn , respectively such that 1 5T . Since
G = <HIJHZJ' ° °2Hn>)
G = (F1,F2, . . .,Fn,Tl,T2, . . ., Tnx

Each of the subsets 13,1.si‘sn, is finite, hence there

exists a finite subset T0 of 61(6) such that

G=<<F F ...,fivT T>

0)

It now follows from the finiteness of T0 and the definition

of ez(G) that there exist subnormal z-subgroups

L1,L2,...,L2 of G such that

<ITO>’C<ZL1,L2,. .. ,le.

 

 

35

Hence,

G = < F1,F2, .. ”Fn’L1” .. ,L£,T\\
If we let

K = <'Fl,. "’Fn’Ll’ .. .,L£\,

an application of Lemma 4.1 yields the existence of a finite

subset L of 91(G) such that

t t t t
91(G)=<F1 ’00.,F ’11 coo

Again, it follows from the definition of 91(G) and the
finiteness of L that there exist subnormal 1-subgroups

M1,M2,...,Mm of G such that

<L>g<M1,M2, . . . ,M > .

m
Hence,

t t t t

—1
935(6) = <1?1 , . . an .

Ll ,. . ”LL ,Ml,...,Mm\t €T\

Since 1 has the property (*), it now follows that

91(G)e:£ . Consequently,
G=<H1,H2,...,Hn>ez;3. D

In the remainder of this section we will point out

a number of interesting consequences of Theorem 4.5.

 

 

36

Definition 4.6: If x is a class of groups, we

 

define the class n01 by

no! = {G‘G is generated by finitely many normal

z-subgroups of G }.

Lemma 4.7 [12; Theorem 1] : Every class x = no}
A
contained in SW8 is a subnormal coalition class. D

Corollary 4.8: The join of finitely many f—subnormal
A A
was — subgroups of G is a 53:8 - subgroup of G . C]

 

Since A\ = n m = sn .A is a subclass
A m :59 0( A39) Om 39)
of {ms, we have as an immediate consequence of Lemma 4.7

and Theorem 4.5

Corollary 4.9: The join of finitely many f—subnormal

 

Mg/xgg-subgroups of a group G is a mg-subgroup of G . D

Remark 4.10: Corollary 4.9 may also be deduced

from Corollary 3.2 of [7].

Lemma 4.ll[ll; Theorem 1]: Every subclass i = n z

 

O
V
of gms is a subnormal coalition class. D
Corollary 4.12: The join of finitely many f-subnormal
V v
5373 - subgroups of G is a {1325 — subgroup of G . D

Remark 4.13: We will give another proof of Corollary

4.12 in Theorem 5.9.

37

The following theorem is proved by S. E. Stonehewer

in [16].

Lemma 4.14 [16; Theorem A]: In any group G, the
join of finitely many subnormal e-subgroups of G is an

6-subgroup of G . D

Theorem 4.5 and Lemma 4.14 yield

Corollary 4.15: In any group G, the join of finitely

 

many f-subnormal 68-subgroups is an 53-subgroup. [3

 

 

Chapter V
V
f-SubnormalfmS—Subgroups

In this chapter we examine the f-subnormal $g-—sub-
groups of a group. In Theorem 5.8 we show that Whenever H
and K are f-subnormal gg-—subgroups, ‘HL:NH(KJ| < m . We
use this result to obtain an alternate proof of Corollary 4.12,
which we give as Theorem 5.9. In Theorem 5.14, we show that
for certain classes of groups the join of two finite f—sub-
normal subgroups is always an f—subnormal subgroup if and only
if the join of two f-subnormal gg-subgroups is always an

f—subnormal subgroup.

V V
Lemma 5.1: If Gems and quqG, then Hgfms

V

Proof: It is clear that K‘smg whenever K<aG

V
and G aim

s

V
Let H;G,G€£ms and |G:H] < co . Then,
V .
[G : CoreG(H)] < an and Coreegi) 651323 . Since H/CoreG(H)€ {5,
v

H/CoreG (H) cams . Hence H 521118138 = mg.

Now, by induction on f(G,H), it follows that every

V
f—subnormal subgroup H of G belongs to fins . [3

38

 

39

V
Lemma 5.2: If Gesms, then G satisfies the minimum

condition for subgroups of finite index. D
V -
Lemma 5.3: If ngHchegms and If
(i) \G:NG(H2)| < a:
and ' 1

(i1) ‘Hz :Hll < co,

 

then ‘Gr:NC(H1)‘ < 9 - I
Proof: If H1 and H2 are as indicated, then
f f
ngHzoNsz) cG . Hence H1 and H2 are f—subnormal sub-
groups of G and it follows from Lemma 5.1 that H1 and H2
v

are mg-—subgroups of G . Let F be the minimum subgroup

of finite index in H2 . F is a characteristic subgroup of

H2 and Fg;Hl . Since H2<3K = NG(H2), F<JK . We con31der

Hl/F gHz/FqK/F .

Since H2/F g3 and HZ/FcK/F, [K/F : NK/F(H1/F)| < m . Hence,

1K: NK(H1)‘ < co and 1G:NK(H1)] < co . It follows that.
‘G:NG(H1)‘<00. C]
v
Lemma 5.4 [11; Theorem 3]: If H<1<1G and Gear-.5
then |G:NG(H)\ < co . D
V

Theorem 5.5: If Gums and H<1f<1G, then

‘G:NG(H)‘<CD. J

 

40

Proof: We prove the assertion by induction on f(G,H).

If f(G,H) = 1, then either H<lG or ‘G:H‘ < co .

In either case the assertion follows trivially.

 

I
Suppose that f(G,H) = n and H = anHn_1¢:-_-...CHI = G g
o ; a
is an f-series for H in G . By the induction hypothesis f .
v 2
‘G':NG(Hn-l)‘ < m . By Lemma 5.1, K = NG(Hn—l)€:fls and we
have i
V
Han—que‘ms .
If H<3Hn_l, then by Lemma 5.4 lKi:NK]H)| < m . If
‘Hn-l :H] < m, then by Lemma 5.3 ‘KZ:NK(H)] < m . Hence

IGerle)| < m, from which we conclude that \G :NG(H)|< m.[j

V
Corollary 5.6: Let Gefims and suppose H and K

 

are f-subnormal subgroups of G . Then J = <fLI<> is an

v
f-subnormalfmS-subgroup of G

‘ggggg: Once we have shown that J<1f<1G, we can con—
clude, using Lemma 5.1, that Jag/ls . Since H<1foG and
GgflES, h3:NG(H)| < o . Proposition 3.11 shows that
J=<H,K><f<lG. Cl

V

° Corollary 5.7: If Gefms and HA<f<IG for leA,

 

then

(i) (W H <1f<JG
leA

 

41

and
(n4 (HflleA><fdG
V
Proof: Since G ems, O N (H ) has finite index in
————— G A
leA
G . Hence, since
0 N (H )s;N ( FIH )
leA G l G AEA l
and

(]N(H)CN(<HIXeA\),
A6AG A G x

the normalizer in G of both NBA and <fgl‘X€:A\‘ has

finite index in G . Hence,

f
F]H <aN FlH ;;G
leA A G\ leA A)
and

f
<Hfll6A>oNGKHAHeA>)§G

are f-series for NBA and <1Hx‘l eA‘\ respectively. The

16A
corollary now follows. E
V
Theorem 5.8: If H and K are f—subnormal mg-—

subgroups of G, then |H :NH(K)‘ < m .

Proof: Suppose that the theorem is false. Then
V
there exists a group G with f—subnormal Tg-—subgroups H
V
and KO such that |H0:NHO(KO)I 4m . Since Hoefils,

O

 

42

there exists a subnormal subgroup H of HO minimal with
_respect to the existence of an f—subnorma1!%S-subgroup K1
of G such that [H : NH(K1)] 4o . Since K1 99335, there
exists a subnormal subgroup K of K1 such that K is
minimal with respect to ‘HI:NH(K)| 4 m . It follows from
Lemma 5.1 that H and K are f-subnormalE;S-subgroups of
G .

Let F be the minimum subgroup of finite index in
H . Since He‘ll/S, F exists and ‘H:F| < on . Suppose
that F EzH, then ‘F’:NF(K)| < m by our choice of H and
K . But NF(K) = FflNH(K) . Hence |H:NH(K)\ < a, con-
tradicting our choice of H and K . Hence H = F . Since
H has no proper subgroups of finite index, H normalizes
every proper subnormal subgroup of K by our choice of H

and K . In particular, H normalizes Q, where Q is

the product of all the proper normal subgroups of K . Thus

Q<aJ = <ILI<>

Let U = HQ/O,V = K/Q, and W = J/Q . Then U and V are
V
f—subnormalEms-subgroups of W . Also, U has no proper

subgroups of finite index and V is simple. An application
of Corollary 2.12 shows that U<aqvv . If V is an infinite
simple group, then V would have no subgroups of finite
index, and consequently would also be subnormal in W . In

V
this case, we can apply Lemma 4.11 to conclude that Wefls

 

43

If V is a finite simple group, then Uv is the join of a

finite number of conjugates of U and, applying Lemma 4.11,

V

we conclude that UV egms and UV<1<1W . In fact, UV<1W
. v v V V
Since W = <U,V> . Hence W/‘U a- V/U flV aims =q£ms . We
V V V
conclude that Wefmsgms =uns.
V V
In either case, W €331 . But W = J/Q and Q 65m
v vS v S
by Lemma 5.1. Hence, Jemsflns =gms . By Theorem 5.5,
|J : NJ(K)] < an . We conclude that |H : NJ(K) flH‘ =
‘H:NH(K)‘<00. D

We now use the results of this section to give

another proof of Corollary 4.12.

Theorem 5.9: If H ,H , . . .,H are f-subnormal
V —— l 2 n V
Ems—subgroups of G then J = <H1,H2, . . . ’Hn> gyms .
Proof: Let Fi’l si sn , be the minimum subgroup
of finite index in Hi . It follows from Lemma 5.2 that Fi
v
exists, Fi guns , and ‘Hi : Fi] < co . By Corollary 2.12,
Fi <1<1G
Let F = <Fi|ls i 5n) . Suppose F has a proper

subgroup L of finite index , then [Fi : FiflL] < a: and
L :Fi,l si sn . Hence L = F . Thus, F has no proper
subgroups of finite index. It follows from Lemma 4.11 that

V
F is a subnormal {ms- subgroup of G . Hence, since F has

 

 

44

no proper subgroups of finite index, F normalizes every

f—subnormal gg-subgroup of G . The subgroup F has
finite index in the group (Hi,F> = HiF , hence F is
the minimum sufigroup of finite index in HiF’ lsnisrl. In
particular, F‘i = F for lusi‘sn. and F<3J .

Now, consider the groups
3 = J/F,Hi = HiF/F, lsisn

The groups Hi are finite f-subnormal subgroups of '3 .

__ V
Hence, it follows from Lemma 2.15 that Je33 . Since F'em%,
V V
J 6513158 = Ems

We obtain the following corollary from the proof of

Theorem 5.9.

 

Corollary 5.10: Suppose H1,H2,. .. ,Hn are f-sub—
v
normal m%-subgroups of G . If Fi and F denote the
minimum subgroup of finite index in Hi and <QHi]1.si‘sn\

respectively, then

F =<Fi\lsisn><1<:G
and

(H1,H2,...,Hn>/Fgg (1

Corollary 5.11: Let H<1<1G and K<1f<1G be such

V
that H and K are fins-subgroups of G . Then <H,K><If<IG.

 

 

45

V
and <(H,K> eflg

Proof: It follows from Theorem 5.8 that \K:NK(H)\<:m.

Thus HK is the join of a finite number of conjugates of H .
Since egg is a subnormal coalition class (Lemma 4.11),
H’Kqu and HKgfl‘I/ls . Lemma 3.3 now implies that

<HK,K> = <H,K> <1qu . It follows from Theorem 5.9 that
<H,K> 6‘ng . D

 

 

Remark 5.12: It follows from Theorem 3.18 and the
V
observation that g is contained in NE that the join of
V
two f—subnormal m%-subgroups of a group G need not be

f-subnormal in G .
Lemma 5.13 [9; Lemma 4.3]: If F is a subnormal
V

ms-subgroup of G such that F has no proper subgroups of

finite index, then s(G,F):;2

 

Proof: If quG, it follows from Theorem 5.8 that

FngG(F) . Hence FGCNG(F) . We conclude that

F<JFG<JG . D

Theorem 5.14: If x =<11 = s: is a class of groups, ]

then the following are equivalent:

(1) If Gegz, the join of two finite f—subnormal

subgroups of G is f-subnormal in G

 

46

V
(2) If G 6!, the join of two f—subnormal Ems - sub-

groups is f-subnormal in G .

 

M: Since {sch/IS , it is clear that property (2)
implies property (1).
V .-
Suppose that H and K are f—subnormal Ems - subgroups
of the I-group G . If we let J = <H,K\, then Jegts
by Theorem 5.9. Let F, FH’ and FK be the minimum subgroups ”-
of finite index in J,H, and K respectively. By Corollary
5.10, F = FH- FK . Lemma 5.13 shows that F<1FG<1G . Hence
F<1FGJ . We now observe that HF/F and KF/F are finite

f-subnormal subgroups of FGJ/F and FGJ/F cf. = s; =q}. .

Hence (1) implies that
< HF/F,KF/F\, = J/F <1 f <1 FGJ/F

Thus J<)f<1FGJ . Also HFG/F‘G and KFG/FG are finite

f-subnormal subgroups of G/FG, where G/FG sq}. = I . Hence,
it follows from (1) that (HFG/FG,KFG/FG> = JFG/FGofos/FG .
Thus, FGchqG . The statements choFGJ and JFG<If<G

imply that J<If<lG . [:1

Corollary 5.15: Let 1 = s: = q! be a class of

 

groups such that every finite subgroup of a 3, -group G is

f—subnormal in G . If G e 391 , then the join of two f-sub-
V V

normal {018 - subgroups of G is an f—subnormal :Ugs - subgroup

of G.

47

Proof: The corollary is a consequence of Corollary

3.7, Theorem 5.14, and Theorem 5.9. D

Remark 5.16: In the statement of Corollary 5.15, we

may take for the class I the classes g,m,m, or PC .

 

Chapter VI

Bf-Groups
The following definition is due to R. Baer [1].

Definition 6.1: We say that G is a nilgroup if for

 

all xeG,<x><1<1G

We generalize the concept of a nilgroup in the fol—

lowing definition.

Definition 6.2: We say that G is a 6f-group if

 

for all xeG,<x><1f<lG

It is clear that every nilgroup is a Bf-group. On
the other hand, since the class 3 is contained in Bf and
not every finite group is a nilgroup, the class Bf properly

contains the class of nilgroups.

Definition 6.3: A non-abelian group in which every

subgroup is normal is a Hamiltonian group.

The structure of Hamiltonian groups is well known.

A description may be found in 9.7.4 of [15].

It is clear that every Hamiltonian group is a Bf-group.

We have the following partial converse.

48

 

49

Proposition 6.4: If G isa non-abelian Bf-group

 

such that for all xeG, f(G,<x>) = 1, then G is a

finitely generated FC—group or a Hamiltonian group.

Proof: If xeG, either <x><1G or [G:<X\[<co.

In either case, [G :CG(x)[ < o . Hence G is an FC—group.

If there is an element x eG such that [G : <x>[ < co,

then G is a finitely generated FC-group and [G :Z(G)[ < o .

 

0n the other hand, if for all x 3G, <x> 4G, then
every subgroup of G is a normal subgroup. In this case, by

Definition 6.3, G is a Hamiltonian group. D

Lemma 6.5 [15; 7.1.7] : If G is a finitely generated
group having a subgroup K such that [G : K[ < on, then

there exists NcharG such that NCK and [G : N[ < co . [:1

Proposition 6.6: Let G be a (Bf—group such that

 

for all x 5G, f(G,<x>) $2 . Then, G/FC(G) is a nilgroup
in which each cyclic subgroup <3?) has subnormal defect

s(G/FC(G),< f>) s2

Proof: Let x eG such that < x> is not subnormal
in G with subnormal defect s(G,<x>)sZ . Then, there
exists a subgroup K of G (K may be equal to G ) such that

either

(a) <X>£K<JG

50

or
f
m) <X><K;G.

AA A
Suppose <X>£K<G. Then KeEUHIP=EDICBQ. By

Lemma 6.5, there exists Nc<x> such that NcharK and

[K : N[ < a: . Hence NqG . Let us consider

<x>/N g K/NdG/N .

Since K/Neg and K/N<1G/N,[G/N:NG/N(<x>/N)[<oo . We

conclude that [G : NG(<x>)[ < a: and hence [G : CG(x)[ < co .

Suppose that <x><1K£G . Then [G:NG(<x>)[ < co .

Since NG(<x3)/CG(<x>) 68: we obtain that [G:CG(<X>)[<oo.

Let N1 = [x€G[<x> has an f—series (a) or (b)]. By
the above remarks, it is clear that the set N1 is contained
in PC (G), the FC—center of G . Since every element of FC(G)
We conclude that N

has an f-series (b), FC(G) cN =FC(G).

l 1

If x(FC(G) =N we must have <x><1xG<1G by

l)
the definition of N1 . Hence, in the group G/FC(G) every
cyclic subgroup (33> is subnormal with subnormal defect

s(G/FC(G),<§>)32 and G/FC(G) isa nilgroup. C]

We need the following commutator notation. We write
[x,ly] for [x,y] and [x,(n+l)y] for [[x,ny],y]. We

write F1(G) for G and Fn+1(G) for [Fn(G),G]

 

51

Definition 6.7: We say that G satisfies the nth

 

Engel condition if [XfilY] = l for all x,ye;G

Groups satisfying the third—Engel condition are

investigated by H. Heineken in [2].

Theorem 6.8 [2; Hauptsatz l and 2]: If G is a

group satisfying the third—Engel condition, then

(i) G is locally nilpotent and
(ii) r5(G) is contained in the direct product of the

Sylow 2-—and Sylow 5-subgroups of G . D

As an immediate consequence of Proposition 6.6 and

Theorem 6.8 we have

Corollary 6.9: Let Ger such that for all XeG,

 

f(G,<J(>):52 . If G/FC(G) has no elements of order two or

five, then G/FC(G)e914 . C]

Definition 6.10: We say that G is locally normal

 

if every finite subset of G is contained in a finite normal

subgroup of G

‘Lemma 6.11 (Dietzmann's lemma) [4; p. 154]: If M
is a finite normal periodic subset of a group G, then <n4>

is a finite normal subgroup of G . D

Lemma 6.12 [15; 15.1.16]: If G is an FC-group

 

52

then G/Z(G) is a periodic FC—group. ‘ E]

Lemma 6.13 [5; Theorem 3.2]: Let G be a locally

A
nilpotent group satisfying T%_. Then G is a finitely
A
generated nilpotent group and hence satisfies 2m . [3

We have the following corollary to Proposition 6.6.

Corollary 6.14: Let G be a Bf-group such that

 

A A
for all xeG,f(G,<x>)52 . If 691%, then Gem).

Proof: Let Z = Z(FC(G)). Since Echar FC(G) charG,

A
Z<1G . Thus, if x€Z,xG;Z . Since Gem)“, there exist
elements Xl’X2" .,xm of '2 such that
— G G G
Z — x1 x2 . xm
Since '2 is abelian and [G :C(xi)[ < m for i = 1,2,.. .,m,

we conclude that El is the direct sum of a finite number of
cyclic groups. Hence Eggs: .

We now consider the subgroup FC(G)/E of G/E .
Since FC(G)/E is periodic by Lemma 6.12, it follows from
Lemma 6.11 that FC(G)/E is finite. It was shown in
Proposition 6.6 that G/FC(G) is locally nilpotent. Since
G/FC (G) 59%;. , Lemma 6.13 implies that G/FC(G) sill?! . Hence

AAA
GQMWM==M. D

Using the results of Chapter V, we have the following

53

proposition.

Proposition 6.15: If G is a Bf-—group satisfying

 

$g], then G is a periodic FC—group with [G :Z(G)[ < m .
_P_£9_o_f: Let xeG . Since Ger,<x><)f<)G . Now,
Lemma 5.1 implies that (X) 5ng . Hence G is a periodic
group. It follows from Theorem 5.5 that [G::NG(<){>fl < m .
Hence [G:CG(x)[ < co for all xcG and GeFC . If F
is the minimum subgroup of finite index in G, FcCG(x) for

all xeG and [G:Z(G)[<oo. D

As consequences of Proposition 3.16 and Proposition

6.15, we have the following corollaries.

 

V
Corollary 6.16: If G is a Bf-group with Ems,
then every subgroup of G is f—subnormal in G . [3
V

 

Corollary 6.17: If G is a wS-group, then every
subgroup of G is f—subnormal in G if and only if

[G:Z(G)[<oo.

Proof: If every subgroup of G is f-subnormal in
V
G, then ngfif/\T% . It follows from Proposition 6.15 that

[G:Z(G)[<ao.
The converse is a special case of Proposition 3.16Jj

Remark 6.18: According to Proposition 3.16, if

54

there exists a positive integer n such that [Gr:Zn(G)| < o,
then every subgroup of G is f-subnormal in G . We leave
as an open question whether the converse of Proposition 3.16

is true.

BIBLIOGRAPHY

 

10.

11.

12.

l3.

14.

Math. 563.

BIBLIOGRAPHY

Baer, R., Nilgruppen, Math. Z., 62 (1955), 402-437.

 

Heineken, Hermann, Engelshe Elemente der Lange drei,
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Kurosh, A. G., The Theory gprroups, Vol. I, New York:
Chelsea, 1960.

 

Kurosh, A. G., The Theory g§_Groups, Vol. II, New York:
Chelsea, 1960.

 

McLain, D. H., 93 locally nilpotent groups, Proc. Cambridge
Philos. Soc., 52 (1956), 5—11.

 

Neumann, Peter M., 93 the structure 9f standard wreath
products of groups, Math. Z., 84 (1964), 343-373.

 

 

Phillips, Richard 3., Some generalizations 2: normal
series in infinite groups. To appear in the J. Austral.

 

 

Robinson, D. J. 8., Infinite Soluble and Nilpotent Groups,
Queen Mary College Math. Notes, London (1967).

Robinson, Derek 8., 92 the theory 9f subnormal subgroups,
Math. Z., 89 (1965), 30-59.

 

Robinson, Derek 5., Joins of subnormal subgroups, Illinois

J. Math., 9 (1965), 144-168.

 

 

Roseblade, J. E., 92 certain subnormal coalition classes,
J. Algebra, 1 (1964), 132-138.

 

Roseblade, James E., A note 9g_subnormal coalition classes,
Math. Z., 90 (1965), 373-375.

 

 

Roseblade, J. E., 92 groups in Which every subgroup i_
subnormal, J. Algebra, 2 (1965), 402-412.

 

Roseblade, J. E. and Stonehewer, S. E., Subjpnctive and
locally coalescent classes 9f groups, J. Algebra, 8 (1968),

 

 

423-435.

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15.

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17.

Scott, W. R., Group Theory, Englewood Cliffs, New Jersey:
Prentice Hall, 1964.

Stonehewer, S. E., The join 2: finitely magy subnormal
subgroups, Bull. London Math. Soc., 2 (1970), 77-82.

 

Wielandt, Helmut, Eine Verallgemeinerung der invarianten
Untergruppen, Math. Z., 45 (1939), 209-244.

 

 

56

IHIIIWIH

l
I