ABSTRACT

THE STRUCTURE OF THE GENERALIZED CENTER
AND HYPERCENTER OF A FINITE GROUP

By

Ram K. Agrawal

Let G be a finite group. Following Ore, a subgroup K of
G is called quasinormal in G if KR = HK for all subgroups H
of G. As a generalization of quasinormality, we say that a subgroup
K of G is n-quasinormal in G if KP 3 PK. for all Sylow subgroups
P of G. Kegel introduced this concept. He proved that a maximal
n-quasinormal subgroup is normal and that a n-quasinormal subgroup is
subnormal. Later, Deskins studied n-quasinormality and proved that
if K is a n-quasinormal subgroup of G, then K/KG is nilpotent,
where KG is the largest normal subgroup of G contained in K.

The aim here is to study n-quasinormality further and to obtain
generalizations of some of the results on normality and quasinormality
by imposing the weaker requirement of n-quasinormality in place of
normality and quasinormality. Throughout, G denotes a finite group.

Gaschfitz has studied the finite groups in which every sub-
normal subgroup is normal while Zacher has studied the groups whose
.subnormal subgroups are quasinormal. They havecharacterized,
resPectively, such finite solvable groups. Extending their results,
we characterize finite solvable groups in which every subnormal sub-

group is n-quasinormal. The main results we obtain in this direction

Ram K. Agrawal

are:

(1) Let G be solvable and D(G) be its hypercommutator
subgroup (the smallest normal subgroup of G such that G/D(G) is
nilpotent). If all subnormal subgroups of G are anuasinormal in
G, then D(G) is a Hall subgroup of odd order and every subgroup of
D(G) is normal in G. In particular, D(G) is abelian.

Note that, since G/D(G) is nilpotent, every subgroup of
G/D(G) is n-quasinormal in G/D(G).

(2) If G (not necessarily solvable) has a normal Hall sub-
group N such that all subnormal subgroups of N are normal in G
and all subnormal subgroups of GIN are n-quasinormal in GIN, then
the subnormal subgroups of G are n-quasinormal in G.

This result implies that the conclusions of (l) are not only
necessary but are also sufficient.

Generalizing the notion of the center of a group G, Ore de-
fined the quasicenter Q(G) of G to be the subgroup generated by
all elements g of G such that «<g> is quasinormal in G.
thherjee has studied the quasicenter in detail and proved that Q(G)
is nilpotent. Like the hypercenter, he also defined and investigated
the hyperquasicenter Q*(G) and proved that Q*(G) is supersolvable.

In an obvious manner, we generalize the above concepts and
the work of Mukherjee. we define the generalized center an(G) of
G to be the characteristic subgroup generated by all elements g
of G such that <g>» is n-quasinormal in G. This leads us to the
definition of the generalized hypercenter. Let (an(G))O = l and
(an(G))i+l/(ZGn(G))i be the generalized center of G/(an(G)>i'

We get an ascending chain of characteristic subgroups:

Ram.Km Agrawal

1 = '(ch(G”o < an<c> -- <an<6>>1 < acnwn2 <...< (zcnmnm =-
* *
ZGn(G)' The terminal member an(G) of this chain is called the
generalized hypercenter of G.
Some of the results we prove here are: (1) Every Sylow sub-

group of Z (G) is generated by the elements g of G such that

Gn
<g> is n-quasinormal in G; (2) an(G) is nilpotent; (3) 2;;(G) =
(KN\N <1G' and ZGn(G/N) = l}; (4) 2;;(6) is supersolvable; (5) G
has the Sylow tower property of supersolvable groups if and only if
G/Z;§(G) has this property; (6) G is supersolvable if and only if
G/Z;n(G) is supersolvable; and (7) 2;;(G) is the product of all
generalized hypercentral subgroups of G, i.e., z;n(c) - <1-I‘H 4G
and for all M <1 G with 14$ H, HIM n an(G/M) + '1'>.

Huppert has studied the structure of a group when its i-th
maximal subgroups (i = 2,3) are normal while Janko has studied the
structure of a solvable group whose 4-th maximal subgroups are normal.
Mann has improved their results by requiring that i-th maximal sub-
groups be quasinormal instead of normal. We further improve these
results and close our present investigation on n-quasinormality.

We prove the results of Huppert, Janko and Mann under the weaker
requirement of n-quasinormality. Our results are:

(1) If every second maximal subgroup of G is n-quasinormal
in G, then G is supersolvable. Furthermore, if \G| is divisible
by at least three different primes, then G is nilpotent.

(2) If every third maximal subgroup of G is "equasinormal
in G, then: (i) if \G‘ is divisible by three or more different
primes, then G is supersolvable; (ii) the commutator subgroup G'

of G is nilpotent; and (iii) the rank of G ' r(G) s 2.

RBULKm Agrawal

(3) Let G be solvable. If every fourth maximal subgroup
of G is n-quasinormal in G, then: (i) if ‘6‘ is divisible by
four or more different primes, then G is supersolvable; (ii)

r(G) s 3.

THE STRUCTURE OF THE GENERALIZED CENTER
AND HYPERCENTER OF A.FINITE GROUP

Bij}

. «3.:
Ram Kf.Agrawal

A DISSERTATION
Submitted to
Michigan State university
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Phthematics

1974

IN MEMORY OF MY MOTHER

ii

ACKNOWLEDGMENTS

I am greatly indebted to Professor WRE. Deskins for his
guidance and many useful suggestions during the preparation and
writing of this thesis. His patience and understanding have played
an important role in the completion of my work. Words are not
adequate to express my deep appreciation and gratitude to him.

I am also grateful to Professor J. Adney and Professor
' I.N. Sinha for their keen interest in my work and for their always
available advice and assistance.

Finally, I take this opportunity to offer my sincere
appreciation and special thanks to my wife, Malti, whose support

and encouragement have made this work possible.

111

TABLE OF CONTENTS

mRODmTIw .00. 000000000000 OOIOOOOOOOOOO ........ 00.0.00...
CHAPTER I. DEFINITIONS AND KNOWN RESULTS ......... ....... ..
1.1 Maximal subgroups, subnormal subgroups, solvable
groups and supersolvable groups ..... ........ ..
1.2 Generalized normality .................. .......
CHAPTER II. FINITE GROUPS WHOSE SUBNORMAL SUBGROUPS ARE
n-QUASINORMAL ..... . ........... ..... ....... ....
2.1 Preliminary results and examples ....... ..... ..
2.2 Solvable (n-q)-groups ..... ........ ............
2.3 (n-q)-groups with special Sylow subgroups .....
CHAPTER III. GENERALIZED CENTER AND HYPERCENTER OF A FINITE
GROUP . ............... . .......... . ...... ......
3.1 Generalized center ........ ....... .. ..... ......
3.2 Generalized hypercenter .. ..... ........... .....
3.3 Generalized hypercentral subgroups ............
CHAPTER IV. FINITE GROUPS WHOSE i-th MAXIMAL SUBGROUPS ARE
n-QUASINORMAL .............. ........
4.1 Definitions and assumed results .. ......... ....
4.2 Generalized results .. ...... .... ...............
BIBLIOGRAPHY ........................ . . . . .......... ..

iv

DUI

12
13
15
21
24
25
32
43
46

47
48

58

INTRODUCTION

Let G be a finite group. ,A subgroup K of G is called
quasinormal in G if K permutes with every subgroup of G, that is,
if KH - HR for all subgroups H of G. This concept was introduced
by Ore [16]. As a generalization of this, we say that a subgroup of
G is n-quasinormal in. G if it permutes with every Sylow subgroup
of G. The n-quasinormal subgroups were first defined and studied
by Kegel [13]. Later, they were also studied by Deskins [5].

It is obvious that a normal or quasinormal subgroup is always
n-quasinormal, but the converse is not true in general. However,
Kegel has shown that a n-quasinormal subgroup is necessarily a sub-
normal subgroup and that a maximal fi-quasinormal subgroup is normal.
Deskins has proved that if K is a n-quasinormal subgroup of G,
then K/K‘.G is nilpotent, where Kt is the largest normal subgroup
of G contained in K.

In the present thesis, we further investigate fl-quasinormality
and generalize a number of results on normality and quasinormality.
Throughout, the groups are finite.

A group G is called a (q)-group ((t)-group) if every sub-
normal subgroup of G is quasinormal (normal) in G. The (q)-groups
and the (t)-groups have been studied by Zacher [18] and Gaschfitz [7]

respectively, who have characterized such finite solvable groups.

A solvable (t)- or (q)-group is always supersolvable.
Gaschfitz has shown that if G is a finite solvable (t)-group and
G/L is the maximal nilpotent factor group of G, then G/L is
Hamdltonian and L is an abelian Hall subgroup of odd order whose
subgroups are normal in G. A similar result was proved by Zacher
for solvable (q)-groups.

As a generalization of (q)-groups and of (t)-groups, we say
that a group G is a (n-q)-group if every subnormal subgroup of G
is n-quasinormal in G. In Chapter II, we study finite (n-q)-groups
and characterize such solvable groups. We extend the above results,
among others, to finite solvable (n-q)-groups and give the conditions
under which a (n-q)-group is either a (q)-group or a (t)-group. ‘We
also give examples of (n-q)-groups that are not (q)-groups.

Generalizing the notion of the center 2(6) of a group G,
Ore [16] defined the quasicenter Q(G) of G to be the subgroup
generated by all elements g of G such that i<g>' is quasinormal
in G. The structure of the quasicenter and its properties have been
studied in detail by Mukherjee [15]. He proved that Q(G) is nilpotent
and that the Sylow subgroups of Q(G) are generated by the elements
g such that 1<g> is quasinormal in G. iMukherjee also extended,
in a natural way, the concept of the hypercenter 2*(G) of G. He
defined and studied the hyperquasicenter Q*(G) of G, and proved
that 1) Q*(G) is supersolvable; 2) Q*(G) - n[N\N <6 and Q(G/N) - l}
and 3) G is supersolvable if and only if G/Q*(G) is supersolvable.

In Chapter III, we generalize the above concepts and the work
of thherjee. We define the generalized center an(G) of‘a group

G to be the subgroup generated by all elements g of G such that

<g> is n-quasinormal in G. Such elements are called the generalized
central elements of G. It is obvious that an(G) is a character-
istic subgroup of G and that if G is nilpotent, then an(G) - G.

The definition of the generalized center leads to the defini-
tion of the generalized hypercenter. It is defined in the same way
as the hypercenter and hyperquasicenter. Let (an(G))O - 1 and
(an(G))i+1/(zcn(c))i be the generalized center of G/(ch(G))1.
This yields an ascending chain of characteristic subgroups:

1 = (ZanD0 < 2611(6) = (ZGn(G))1 < (ZGn(G))2 <---< (ZGnKD)In =
z;n(c). The terminal member Z;g(G) of this chain is called the
generalized hypercenter of G.

Some of the results we prove here are: 1) If <g>' is
n-quasinormal in G, then <gr>r is also n-quasinormal in G for all
integers r; 2) Every Sylow subgroup of an(G) is generated by
the generalized central elements of G; 3) an(G) is nilpotent;

z.) 2211(9) = nmn <1 G and zcnmm - i}; 5) c is supersolvable
if and only if G/Z;;(G) is supersolvable; 6) G has the Sylow
tower property of supersolvable groups if and only if G/Z;g(G) has
the same prOperty; and 7) 2;;(6) is supersolvable.

A number of mathematicians have studied the structure of the
group when its i-th maximal subgroups satisfy some imbedding property.
In this direction, Huppert [9], Janko [12] and Mann [14] have proved
the following theorems for a finite group G.

(Huppert). If each second maximal subgroup of G is normal
in G, then G is supersolvable. If the order of. G is divisible

by at least three different primes, then G is nilpotent.

(Huppert). Let each third maximal subgroup of G be normal
in G. Then: (i) the commutator subgroup G' is nilpotent; (ii)
the rank of c = r(G) s 2; (iii) if ‘G| is divisible by at least
three different primes, then G is supersolvable.

(Janka). Let G be solvable. If each fourth maximal sub-
group of G is normal in G, then: (i) r(G) s 3; (ii) if \G\
is divisible by at least four different primes, then G is super-
solvable.

(Mann). Let G be solvable and each n-th maximal subgroup
of G be quasinormal in G. Then: (i) r(G) s n-l; (ii) if \G‘
is divisible by at least n-k+l distinct primes, then r(G) s k,
where k 2 1.

we conclude our investigation by improving the above results.
In Chapter IV, we prove these results under the weaker assumption of
n-quasinormality instead of normality or quasinormality.

For the sake of completeness, we have collected some basic

definitions and known results in Chapter I.

CHAPTER I -

DEFINITIONS AND KNOWN RESULTS

This chapter contains, for easy reference, some basic defini-
tions and results which are frequently used throughout the present
work. Some proofs are also included. The results without references

can be found in Huppert's book [8]. All groug considered here are

finite.

1.1 Maximal subgroups, subnormal subgpoups, solvable groups and

 

supersolvable groups.
'Definition 1.1.1: A subgroup H of a group G is maximal

in G if there is no proper subgroup K of G such that H $K.

1<...<Hl<Ho-G of G isamaximal

chain if each H1 is maximal in Hi-l for i = 1,2,...,n. A sub-

group H of G is an n-th maximal subgroup of G if there is at

A chain of subgroups fin < Hn-

<...<H <H IG

least one maximal chain of subgroups Hn < Hn-l 1 O

with Hn = H.

It should be noted that an n-th maximl subgroup of G my
also be an m-th maximal subgroup of G for m 1‘ n. For example,
consider $4, the symmetric group on four letters, and if H is a
Sylow 2-subgroup of 83 in 84, then H is a second as well as a
third maximal subgroup of 84.

Theorep_1L1_.2_: If H is a maximal subgroup of a solvable
group G, then [G:H] is a power of a prime.

5

Definition 1.1.3: A subgroup H of a group G is subnormal

in G if there is a chain of subgroups: G 3 Go l> G1

such that G1 is normal in‘ G1_1 for i - 1,2,...,n.

It is well-known that if H and K are two subnormal sub-

DGZ D...>G -H
n

groups of G, then <H,K> and H n K are again subnormal subgroups
of G. It follows from the definition that if H is subnormal in
G and K is a subgroup of G, then H n K is subnormal in K and
if H s K, then H is subnormal in K.

Definition 1.1.4: For a subgroup H of a group G, the cpr_e_

of H in G, denoted by H is the largest normal subgroup of G

G,
contained in H, and the subnormal core of H in G, denoted by

H is the largest subnormal subgroup of G contained in H.

56’

The subnormal core of a subgroup need not be the core of the
subgroup. For example, let G - A4, the alternating group of degree
4, and H be a subgroup of order 2. Since H is subnormal in G,
H =H. ButH isnotnormalinGandso H-lI‘HS

SG G
for maximal subgroups, equality does hold.

G . However ,

Lemma 1.1.5: If M is a maximal subgroup of G, then
"so 3 MG'

Proof: Clearly, “G s "56' To prove "SC 5 MG, it is enough
to show that "SC is normal in G.

If M is normal in G, then MSG .- MG and hence ”SC 5 NC.
On the other hand, if M is not normal in G, then we consider the
subnormal chain: “SC = N0 $ N1 $N2 g...$ Nk -- G. Since N1 is
subnormal in G and MSG$N1, it follows that N1 £ M. Hence

Gad M>. Note that N1 normalizes MSG. We will show that M

1’
also normalizes "36' Let xEM. Then {11ka is subnormal in

G. Since x-IMEGx s x-lMx I M, it follows that x-1M§Gx s MSG' But
|x 1MLSGx| I ‘MSc' and so x-IMSGx I MSG' Hence M normalizes MCG'
This means that MSG is normal in G. Therefore, MSG s Ht.

We now turn our attention to the results on the Sylow sub-
groups and Hall subgroups. ‘We begin with a lemma known as the
Frattini lemma.

Lemmp 1.1.6: Let H be a normal subgroup of G. If P is
a Sylow subgroup of H, then G I HNG(P), where NG(P) is the
normalizer of P in G.

Using the fact that the Sylow subgroups, for a given prime,
of a group G are conjugate in G, we prove the following:

Proposition 1.1.7: A subnormal Sylow subgroup of G is
characteristic in G.

2592;: It suffices to show that a subnormal Sylow'subgroup
P of G is normal in G. For this, we may assume without loss of
generality that P g K g G. Clearly, P is a Sylow subgroup of K.
Let g E G. Then g-lKg = K. and so g-ng is a Sylow subgroup of
K. But P is normal in K. Hence g'ng I P, which implies that
P is normal in G.

The next result, due to P. Hall, is a generalization of the
Sylow theorems that holds in finite solvable groups.

Theorem 1.1.8: Let G be a solvable group of order ab,
where (a,b) = 1. Then G contains at least one subgroup of order
a, and any two such subgroups are conjugate. [

Definition 1.1.9: Let \G‘ I ppa, where (pp,a) I 1. A sub-

group of G of order a is called a p-complement of G.

Hall's theorem tells us that every finite solvable group has
a p-complement for every prime p.

Definition 1.1.10: A subgroup H of a group G is called
a Hall subgroup of G if (\H|, [G:H]) = l.

The following result which holds for Hall subgroups of a
solvable group is an extension of Proposition 1.1.7.

Proposition 1.1.11: A subnormal Hall subgroup of a solvable
group is characteristic.

Definition 1.1.12: A group G is called supersolvable if

 

every homomorphic image of G contains a cyclic normal subgroup.

The subgroups and the factor groups of supersolvable groups
are again supersolvable. An extension ofia supersolvable group by
a supersolvable group is not necessarily supersolvable. However,
an extension of a cyclic group by a supersolvable group is super-
3 olvab le .

A remarkable characterization of supersolvable groups was
obtained by Huppert [9].

Theorem 1.1.13: A group G is supersolvable if and only if
every maximal subgroup of G is of prime index.

Definition 1.1.14: The commutator subgroup G' of a group
G is the subgroup generated by all commutators x-ly-lxy for all
x, y in G.

Theorem 1.1.15: The commutator subgroup of a supersolvable
group is nilpotent.

Let H and K be two subgroups of G. Then [H,K] denotes

the Subgroup 1<h-1k-1hk\h e H, k 6 K>u Let y1(G) I G and

Yi+1(G) = [v1(G), G]. This yields a chain of normal subgroups of G.
Definition 1.1.16: The lower central series of G is the

normal series

G = y1(G) 2 y2(G) 2 ...

The terminal member of this series is called the hypercommutator
subgroup D(G) of G.

The above definition of the hypercommutator is equivalent
to the following:

Definition 1.1.17: The hypercommutator subgroup D(G) of
a group G is the intersection of all normal subgroups N of G
such that G/N is nilpotent. That is, D(G) is the smallest normal

subgroup of G such that G/D(G) is nilpotent.

1.2 Generalized Normality
Definition 1.2.1: Two subgroups H. and K of a group G
pgrmute if <H,K> = HK.= KB. A subgroup H of G is guasinormal
in G if H permutes with every subgroup of G.
The concept of a quasinormal subgroup is a generalization of
the notion of a normal subgroup, which was introduced by Ore [16]
and later studied by Iwasawa [11], 1:8 and Szép [10] and Deskins [5].
Clearly, a normal subgroup is always a quasinormal subgroup.
But the converse is not, in general, true as shown below by an
example. However, Ore proved that a quasinormal subgroup is neces-

sarily a subnormal subgroup.

2
Example: Let G = <x,y\xp = yp = 1, y-lxy = xp+l>5 where

p is an odd prime. Then <y> is a quasinormal subgroup of G

which is not normal in G.

10

As a generalization of quasinormality, we have

Definition 1.2.2: A subgroup of a group G is called
n-quasinonmal in G if it permutes with.every Sylow subgroup of G.

The above concept was first introduced and studied by Kegel
[13]. It was later studied by Deskins [5]. Note that a quasinormal
subgroup is always a n-quasinormal subgroup, but the converse is not
necessarily true.

Let us point out here that every subgroup of a nilpotent group
is n-quasinormal in the group. This is so because all the Sylow sub-
groups of a nilpotent group are normal in the group. It is easy to
see that there are nilpotent groups in which every subgroup is not
quasinormal. This confirms that a n-quasinormal subgroup is not
always a quas inormal subgroup.

The following results describe some of the basic properties
of n-quasinormal subgroups.

Lemma 1.2.3: If H is n-quasinormal in G and e is a

9 is n-quasinormal in Ge. In particular,

homomorphism of G, then H
H/N is n-quasinormal in G/N for all normal subgroups N of G
contained in H.

M: The lemnn follows from the fact that the Sylow sub-
groups of G6 are the images of the Sylow subgroups of G under a
and the fact that the property of permutability of subgroups is pre-
served by 9.

Lemma 1.2.4: Let N 5.H s G and N be normal in G. If
H/N is n-quasinormal in GIN, then H is n-quasinormal in G.

Proof: Let P be a Sylow subgroup of G. Since PNIN is

a Sylow subgroup of GIN and HIN is n-quasinormal in GIN, we

11

have (HIN)(PNIN) = (PNIN)(HIN). Therefore, for h in H and k

in P, there exist some h0 in H and k in P such that

0
(hN) (kN) I (kON) (hON) . Hence hkN = kohON, which implies that
halkglhk belongs to N. This means that halkalhk belongs to H
since N is contained in H. Therefore, there is some h in H

1

such that h-lk-lhk = h and so hk = ko(h0h1). From this, it

0 O 1
follows that HP = PH. This proves that H is n-quasinormal in G.
The results in the next theorem are due to Kegel [13].
Theorem 1.2.5: For a group G, we have:
(i) If H s K s G and H is n—quasinormal in G, then
H is also n-quasinormal in K.
(ii) If H is a maximal n-quasinormal subgroup of G, then
H is normal in G.
(iii) If H is a n-quasinormal subgroup of G, then H is
subnormal in G.
Theorgmgllfi (Deskins [5]). If K is a n-quasinormal

subgroup of G and KC is the core of K in G, then KIKG is

nilpotent .

CHAPTER II

FINITE GROUPS WHOSE SUBNORMAL
SUBGROUPS ARE n-QUASINORMAL

We recall from Chapter I the fact that a n-quasinormal sub-
group of a group is always subnormal in the group. Since a subnormal
subgroup of a group is not necessarily n-quasinormal in the group,
it seems natural to ask: "What can be said about the structure of a
group if all of its subnormal subgroups are n-quasinormal in the
group?" We call such a group a (n-q)-group and, in this chapter, we
study finite (fl-q)-groups. Especially we study and characterize,
in Section 2, finite solvable (n-q)-groups. Theorems 2.2.3, 2.2.4,
and 2.2.5 establish the characterization.

A group G is called a (q)-group ((t)-group) if every sub-
normal subgroup of G is quasinormal (normal) in G. The (q)-groups
and the (t)-groups have been studied by Zacher [18] and Gaschfitz [7]
reapectively, and they have characterized such finite solvable groups.

A solvable (t)- or (q)—group is always supersolvable. The
main result of Gaschfitz states that if G is a finite solvable (t)-
group and if D(G) is its hypercommutator subgroup, then GID(G)
is Hamiltonian (a group in which every subgroup is normal) and D(G)
is an abelian Hall subgroup of odd order whose subgroups are normal
in G. A similar result was proved by Zacher for the solvable (q)-

groups.

12

13

We extend the above results, among others, to finite solvable
(n-q)-groups and give the conditions under which a (n-q)-group is
either a (q)-group or a (t)-group. We also give examples of (fl-q)-

groups that are not (q)-groups.

2.1 Preliminary Results and Examples

Definition 2.1.1: A group is a (fl-q)-group if all of its
subnormal subgroups are n-quasinormal in the group.

Remark: In view of Theorem 1.2.5(i), the above definition
is equivalent to the following definition:

A (n-q)-group is a group in which n-quasinormality is a tran-
sitive relation.

The next two inheritance properties of (fl-q)-groups are
immediate consequences of Theorem 1.2.5(i) and Lemma 1.2.3.

(2.1.2) A subnormal subgroup of a (n-q)-group is again a
(n-q)-erUP-

Note that a non-subnormal subgroup of a (n-q)-group is not
necessarily a (n-q)-group, as confirmed by the following example:

Let G I A the alternating group of degree 5, and let H denote

5’
A4 in 1A5. Since A5

group. But H is not a (n-q)-group because its subnormal subgroups

is simple, it follows that A5 is a (n-q)-

of order 2 do not permute with the Sylow 3-subgroups of H.
We will show later that if G is a solvable (n-q)-group,
then all subgroups of G are (n-q)-groups.

(2.1.3) The factor groups of a (n-q)-group are again

(n-q)-grOUP8-

14

12925.3 Let G be a (n-q) -group and N be a normal sub-
group of G. Suppose HIN is a subnormal subgroup of GIN. Then
H is subnormal in G and so H is n-quasinomal in G. Hence
by Leuma 1.2.3, HIN is n-quasinormal in GIN, which implies that
GIN is a (n-q) -group.

Proposition 2.1.4: If G and G2 are two (Tl-q) -groups

 

1
and (\GI‘ , \G2\) = 1, then G I G1 X G2 is a (n-q)-group.
Proof: Let H be a subnormal subgroup of G I G1 X G2 and

P be a Sylow p-subgroup of G. To prove that G is a (Tr-q) -group,
we must show that H and P permute. Since H is subnormal in

G, it follows that for every Sylow subgroup S of G, H n S is a
Sylow subgroup of H. This, together with (‘G1‘, ‘GZD I 1, implies
that H I (H n GI) x (H n Gz) . Clearly, we may assume without loss

of generality that P s G Since H n G1 is subnormal in G and

1' 1
G1 is a (n-q) -group, H ['1 G1 is n-quasinormal in G1. Hence H ['1 G1
permutes with P. lbreover, H [‘1 G2 centralizes (a fortiori, permutes
with) P and, therefore, (H 0 G1) x (H 0G2) I H permutes with P.
This proves the proposition.

1311533: In the above proposition, the condition that

QC“, 162‘) = 1 cannot be omitted. The following example shows

this.
Let G1 = S3 I <x,y\x3 I y2 I 1, yx I x2y> and

G2 I <z\z3 = 1>. Then <xz> is subnormal in G1 X G2. But «2)

is not n-quasinormal in G1 X‘G2 since it does not permute with the

Sylow 2-subgroup (y) of G1 XGZ.
As noted in Chapter I, every subgroup of a nilpotent group

is n-quasinormal but not necessarily quasinormal. Since all subgroups

15

of a nilpotent group are subnormal, it follows that a nilpotent
group is always a (n-q)-group but not always a (q)-group. The follow-
ing examples show that there also exist non-nilpotent groups which
are (n-q)~groups but not (q)-groups.
Example 1. Let p be a prime greater than 3. Denote by

2
- +
G the direct product of the groups <a,b‘ap I bp I l, b 1ab I a1 p>

l
and <c\cp2 I l>u Let G I G1 X 83, where S3 is the symmetric group
on 3 letters. Then G is a (n-q)-group by Proposition 2.1.4. But
G is not a (q)-group since its subnormal subgroup <b>r does not
permute with <ac>u Note that G is solvable but not nilpotent.
Example 2. Let H be a non-abelian simple group and K be
a nilpotent group that is not a (q)-group. Let (‘H\, |K|) I 1.
Such K certainly exists (for instance, G1 of last example when
p > \H|). Now H x K is a non-solvable (n-q)-group which is not a
(q)-sroup-
339355; Note that a (q)-group is always a (neq)-group.

Hence it follows that the class of (n-q)-groups is larger than the

class of (q)-groups defined by Zacher in [18].

2.2 Solvable (n q)1groups

Here we characterize the solvable (n-q)—groups. We begin
with the following observation.

Lemma 2.2.1: Let G be a (n-q)-group. If N is a solvable
minimal normal subgroup of G, then the order of N is a prime.

Ppppf: Since a characteristic subgroup of a normal subgroup
is normal in the group and since N is a minimal normal subgroup,

it follows that N does not contain any proper characteristic

16

subgroups. By the solvability of N, the commutator subgroup N'

of N is a proper characteristic subgroup of N. Hence N' I 1
and so N is abelian. Now, if p is a prime divisor of \N‘, then
the Sylow p-subgroup Np of N is characteristic in N. Since

Np I 1, it follows that N ailt‘Np. Hence N is a p-subgroup. This
means that every subgroup of N (being subnormal in G) is fl-quasi-
normal in G.

Let P be any Sylow p-subgroup of G. Then N is a normal
subgroup of P and N r\Z(P) I 1, where Z(P) is the center of P.
Let g be a non-identity element of N rlZ(P). Since <g>' is
n-quasinormal in G, it follows that <g> is subnormal in <g>Q I
Q<g> for all Sylow q-subgroups Q of G for primes q I p. From
this and the fact that <g>’ is a Sylow p-subgroup of <g>Q, we obtain
that <g>- is normal in <g>Q for all Q, q I p. But <g>) is normal
in P and hence <g>. is normal in G. Since N is a minimal normal
subgroup of G, we have N I <3), which implies that ‘N‘ I p, as
desired.

As mentioned earlier, Zacher [18] has shown that a solvable
(q)-group is supersolvable. Using the above lemma, we prove the same
for a solvable (n-q)-group.

Theorem 2.2.2: A solvable (n-q)-group G is supersolvable.

Ppppf; we use induction on the order of G. Let N be a
minimal normal subgroup of G. Then N is solvable and hence
\N| I p, a prime. By (2.1.3), GIN is a solvable (n-q)-group. There-
fore, GIN is supersolvable by induction. Since N is cyclic, it

follows that G itself is supersolvable.

17

Our next theorem generalizes the main result, stated in the
introduction, of Gaschfitz [7] and the corresponding result of Zacher [18].

Theorem 2.2.3: Let G be a solvable (n-q)-group and D(G)
be the hypercommutator subgroup of G. Then

(i) D(G) is a Hall subgroup of G of odd order, and

(ii) every subgroup of D(G) is normal in G.
In particular, D(G) is an abelian subgroup of G.

33:33: Note that if G is a (t)-group, then GID(G) is
a nilpotent (t)-group and hence every subgroup of GID(G) is normal
in GID(G). In our case, the subgroups of GID(G) are n-quasinormal
in GID(G) and that is what we should expect since we have replaced
normality by n-quasinormality. Also note that every complement of
D(G) in G is nilpotent.

Proof of Theorem 2.2.3: We proceed by induction on the order
of G. Let p be the largest prime divisor of ‘G‘ and P be a
Sylow p-subgroup of G. Since G is supersolvable by Theorem 2.2.2,
P is normal in G. Also D(G) is contained in the q-complement of
G for the smallest prime divisor q of \G‘ because G is q-
nilpotent. Hence the order of D(G) is odd. we now have two cases
according to whether or not p divides the order of D(G).

Case 1. p does not divide ‘D(G)\. Then P (\D(G) I 1 and
so P centralizes D(G). By induction, D(GIP) I D(G)PIP is a Hall
subgroup of GIP and every subgroup of D(G)PIP is normal in GIP.
This means that D(G), since (\P‘, \D(G)‘) I 1, is a Hall subgroup
of G and, for H s D(G), HP is normal in G. Since H is centralized
by P, H is a normal Hall (hence characteristic) subgroup of HP,

which implies that H is normal in G.

18

Case 2. p divides \D(G)‘. We will show that P s D(G).
Let a be an element of P. Since <a> is subnormal in G, <a>
is n-quasinormal in G. Let K be a p-complement of G. Then
<a>K I K<a> is a subgroup. Since <a>» is a subnormal Sylow p-sub-
group of <a>K, <a> is normal in <a>K. But a is an arbitrary
element of P and so every subgroup of P is normalized by K.
Hence every element of K induces a power automorphism in the
elementary abelian group PI§(P) of order prime to p, where Q(P)
is the Frattini subgroup of P. Thus, for every k in K, there

exists a positive integer m(k) such that ak I am(k)

mod Q(P) for
all a in P. Let Yh(c) be the terminal member of the lower
central series of G. Then Yn(G) I D(G). Since p divides \D(G)‘,
K does not centralize P. For, otherwise K would be normal in G
and so GIK would be nilpotent, which would imply that D(G) S K,
an impossibility. Hence there is some y in K which does not
centralize P. Now it follows from Theorem 11.7 in [17] that
m(y) I 1 mod p and so, for all x in P - Q(P), the commutator
((...((x,y1),y2),...),yn) I x<m<¥)"1)n ‘ 1 mod §(P), where
y1 I y2 I...I yn I y. This says that if AI§(P) SiPI§(P) and
\A/§(P)\ I p, then Yh(G) contains an element g such that
g E P - Q(P) and gQ(P) generates A/§(P). The Burnside Basis
Theorem yields that P s yn(P) - D(G). By induction, D(GIP) -
D(G)PIP = D(G)/P is a Hall subgroup of ale, which implies that
D(G) is a Hall subgroup of G.

Next we prove that every subgroup of D(G) is normal in G.

Clearly, we need only show that every cyclic subgroup of D(G) is

normal in G. Let <c>’ be any cyclic subgroup of D(G). Then

19

c I uv I vu for some p-element u and p'-e1ement v and <c> =
<u> x <N>u Since G is supersolvable, its commutator subgroup G'
is nilpotent. But D(G) s.G' and so D(G) is nilpotent. Hence all
subgroups of D(G), in particular <a>, and <y>3 are n-quasinormal
in G. It now follows that <u>r is a subnormal Sylow p~subgroup
of <M>Q I Q<u> for all Sylow q-subgroups Q of G and primes

q I p. Therefore, <u>» is normal in <n>Q for all Q and primes
q I p and so <n>- is normal in G[p], where G[p] is the normal
subgroup of G generated by all p'-elmments of G. Since GIG[p]
is nilpotent, D(G) s.G[p]. This, together with P 5'D(G), implies
that G[p] I G and so <u> is normal in G. Since <y>PIP is a
subgroup of D(G)/P I D(G/P), we have by induction that <N>PIP is
normal in GIP. Thus <v>P is normal in G. But <v>1> is nil-
potent. Hence <v> is characteristic in <y>P and so .<v> is
normal in G. Therefore, <c> is normal in G and this takes care
of case 2.

Finally, note that D(G) is Hamiltonian of odd order. It
is well-known that such a group is always abelian. Hence D(G) is
abelian and the proof of the theorem is complete.

The next theorem gives sufficient conditions for a group G
to be a (n-q)-group. Here we do not require that G be solvable.

Theorepp2.2.4: Let the group C have a normal Hall sub-
group N such that

(i) GIN is a (n-q)-group, and

(ii) every subnormal subgroup of N is normal in G.

Then G is a (n-q)-group.

20

2319;: Let H be a subnormal subgroup of G. Then we must
show that H is n-quasinormal in G. Let N n H I 1. Since N n H
is subnormal in N, N (\H is normal in G by (ii). Now consider
the factor group GIN n H. By induction, HIN n H is n-quasinormal
in GIN n H. It follows from Lemna 1.2.4 that H is n-quasinormal
in G.

Next suppose that N n H I 1. By the improved version of
Schur and Zassenhaus theorem (after the well-known theorem of Feit
and Thompson), G splits over N and all complements of N in G
are conjugate. Let M. be any complement of N in G. Then M,
being isomorphic to GIN, is a (n-q)-group. Since H is subnormal
in G and (\N], \M\) I 1, it follows, as in PrOposition 2.1.4,
that H I (H nM)(H nN). But H nN I 1 and so H IH {11%. This
means that H s M. Hence every complement of N is a (n-q)-group
and contains H. Note that (\H‘, ‘N|) I 1.

Now consider the subgroup HN. Then H is a subnormal Hall
subgroup of HN. We may assume without loss of generality that
H 4 K 4 HN. Since H is a normal Hall subgroup of K, H is the
unique subgroup of K of order ‘H‘, which implies that H is
normal in HN. This means that H permutes with every Sylow sub-
group of’ N. Let p be a prime divisor of the order of G and Gp
be a Sylow p-subgroup of G. If p divides the order of N, then
Gp s N and, therefore, HGp I GPH. On the other hand, if p does
not divide the order of N, then there is a complement L of N in
G such that Gp 5L. Since H is a subnormal subgroup of L and
L is a (an)-group, the subgroups H and GI) must permute. Hence

H is n-quasinormal in G. This proves the theorem.

21

From this we obtain the following result and see that the
conditions (i) and (ii) of Theorem 2.2.3 are not only necessary but
are also sufficient.

Theorem 2.2.5: Let G have a normal Hall subgroup N such
that

(i) G/N is a solvable (rt-q) -group, and

(ii) N is solvable and all of its subnormal subgroups are

normal in G.
Then G is a solvable (n-q)-group.

11592;: Since GIN and N are both solvable, it follows
that G is solvable. The rest is obvious from Theorem 2.2.4.

W: The condition (i) of the above theorem is auto-
matically satisfied if the factor group GIN is nilpotent.

Corollary 2.2.6: Let G be a solvable (n-q)-group. Then
all subgroups of G are again solvable (n-q) ~groups.

P_r_pp_f_: Let K be any subgroup of G and consider K flD(G).
It follows from Theorem 2.2.3 that K n D(G) is a normal Hall sub-
group of K and its subnormal subgroups are normal in K. But
D(G)K/D(G) I K/K fl D(G) and hence KIK n D(G) is nilpotent. Now

K is a solvable (n-q) -group by Theorem 2.2.5.

2.3 (n-g)-groups With Smcial Sylow Subgoups

In this section, we obtain conditions for a (rt-q) -group, not
necessarily solvable, to be a (q) -group or a (t) -group. We need the
following definition.

Definition 2.3.1: A group G is called quasi-Hamiltonian

if all of its subgroups are quasinormal in G.

22

Iwasawa [11] has shown the existence of quasi-Hamiltonian
p-groups that are not Hamiltonian. This suggests the next theorem.

Theorqp 2.3.2: Let G be a (n-q)-group. If all of its
Sylow subgroups are quasi-Hamiltonian, then G is a (q)-group.

2322;; Let K be a subnormal subgroup of G. Then we must
show that K permutes with every subgroup of G. Since the factor
groups of G satisfy the conditions of the theorem, it is sufficient
to consider the case when K

G‘
K is n-quasinormal in G, it follows from Theorem 1.2.6 that K is

the core of K. in G, is <l>. Since

nilpotent. Hence every subgroup of K, which is subnormal in G, is
n-quasinormal in G. Let p be a prime divisor of the order of K.
Then, since the Sylow p-subgroup Kp of K is n-quasinormal in G,
Kp is a subnormal and hence normal Sylow p-subgroup of the subgroup
KHGQ I GqKp for all Sylow q-subgroups Gq and primes q I p. There-
fore, Kp is normalized by every p'-element of G. Since KP permutes
with every Sylow p-subgroup of G, it follows that Kp is contained*
in every Sylow p-subgroup of G. But the Sylow subgroups of G are
quasi-Hamiltonian. Hence Kp permutes with every p-subgroup of G.
Now let g be any element of G. Then <g>I <u> X <v> for some
p—element u and p'-e1ement v, and so <g>- and K.P permute. Since
K is the direct product of its Sylow subgroups, it follows that K
and <g>l permute, which implies that K permutes with every sub-
group of G. Hence K is quasinormal in G. This completes the
proof.

Theorem 2.3.3: Let G be a (n-q)-group. If all of its Sylow

subgroups are Hamiltonian, then G is a (t)-group.

23

2522;; Let K be a subnormal subgroup of G. Duplicating
the above argument, we see that Kp is normalized by every p'-e1ement
of G and, since the Sylow subgroups of G are Hamiltonian, K.P is
a normal subgroup of every Sylow p-subgroup of G. Hence Kp is

normal in G, which implies that K is normal in G.

CHAPTER III

GENERALIZED CENTER AND HYPERCENTER
OF A FINITE GROUP

Generalizing the notion of the center Z(G) of a group G,
Ore [16] defined the quasicenter Q(G) of G to be the subgroup
generated by all elements g of G for which. <g>' is quasinormal
in G. The structure of the quasicenter and its properties have been
studied in detail by Mukherjee [15]. He proved that Q(G) is nil-
potent and that the Sylow subgroups of Q(G) are generated by the
elements g such that <g>: is quasinormal in G. Mukherjee [15]
also defined and studied the hyperquasicenter Q*(G) of G which is
a generalization of the concept of the hypercenter 2*(G) of G.
Some of the results he proved are: (1) Q*(G) is supersolvable;

(2) q*(c) = nmu <1 G and Q(GIN) - i}; (3) 1r 1: 4c and
T s.Q*(G), then Q*(GIT) I Q*(G)IT; and (4) G is supersolvable if
and only if G/Q*(G) is supersolvable.

In this chapter, we generalize the above concepts. We define
and investigate another center and hypercenter of a finite group, and
extend Mukherjee's results.

We define the generalized center an(G) of a group G to
be the subgroup generated by all elements g of G for which <g>
is n-quasinormal in G. In a natural way, the definition of the

generalized center leads to the definition of the generalized

24

25

*
hypercenter ZGn(G)- Let (ZGn(G))O ‘ 1 and (ZGn(G))i+1

be the generalized center of G/(an(G))i' From this we get an

I<zcn(c)> i

ascending chain of characteristic subgroups: l I (ZGn(G))O‘< ZGn(G) I
(an(G))1 < (ZGn(G))24<..a< (an(G))m ‘ Z;n(G). The terminal member
z;n(s) of this chain is called the generalized hypercenter of G.

The main results proved here are: (1) If <g>» is “Iquasi-
normal in G, then every subgroup of <g>' is again n-quasinormal
in G; (2) Every Sylow subgroup of ZGn(G) is generated by the
elements g of G such that 4<g> is n-quasinormal in G; (3) an(G)
is nilpotent; (4) If G I H X K, then an(G) I an(H) X ZGn(K) and
zznm) - zgnm) x 2;,(193 (5) zznm) =- n{n\N ac and an(c/N) - i};
(6) If T as and 1: s zgnm), then zznml'r) -= z;n(c)/'r; (7) zznm)
is supersolvable; (8) G is supersolvable if and only if GIZS;(G)
is supersolvable; (9) G has the Sylow tower property of super-
solvable groups if and only if G/Z;#(G) has the same property;
(10) z;n(c) is contained in the intersection of the maximal super-
solvable subgroups of G; and (11) 2;;(G) is the product of all
generalized hypercentral subgroups of G (these subgroups are defined

in section 3.3).

3.1 Generaliged Center

The aim here is to generalize the notions of the center and
of the quasicenter of a group G to that of the generalized center of
G and to obtain some generalizations of thherjee's results. We
begin with the following simple observation.

Lemma 3.1.1: Let p be a prime divisor of the order of G

and H be a p-subgroup of G. If H is “Iquasinormal in G, then

26

H is normal in HGq for every Sylow q-subgroup Gq of G and
primes q I p.

Ppppf: By Theorem 1.2.5(iii), H is subnormal in G. Since
H permutes with GH’ HGq is a subgroup of G. Hence H is sub-
normal in HGq. But H is a p-subgroup and so H is a subnormal
Sylow p-subgroup of HGq. By Proposition 1.1.7, H is normal in HGq.

Definition 3.1.2: The generalized center ZGn(G) of a group
G is the subgroup generated by all elements g of G such that
<g> is n-quasinormal in G. Such elements shall be called the
generalized central elements of G.

Clearly, an(G) is a characteristic subgroup of G which con-
tains the quasicenter Q(G) and the center Z(G) of G. we shall
show later, by way of an example, that Q(G) can be a proper subgroup
of ZGn(G).

Remark: Note that if G is nilpotent, then an(G) I G.

This is so because every subgroup of a nilpotent group is n-quasinormal.

It is a simple fact that the subgroups of a cyclic normal sub-
group of a group are again normal in the group. In the following
theorem, we prove a similar result for the cyclic n-quasinormal sub-
groups.

Theorem 3.lg3: If <g>’ is n-quasinormal in G, then every
subgroup of <g>' is also n-quasinormal in G.

Ppppf: Let <gn>' be a subgroup of 4<g>, where n is an
integer. To prove the theorem, we must show that <<gn>Gp I s§<gn>
for an arbitrary but fixed Sylow p-subgroup Gp of G. By Theorem
1.2.5(iii), <g> is subnormal in G. Hence <3“), is subnormal in

G. Let P be the Sylow p-subgroup of <3“), and K be the

27

p-complement of <gn>. Since P is a subnormal p-subgroup of G,
it follows that P is contained in every Sylow p-subgroup of G.
Hence P s Cp and so PGp I GDP I GP. Let Q be the Sylow q-sub-
group of <g> for prime q I p. Since <g> is subnormal in C,

Q is subnormal in G. This means that Q is a subnormal Sylow
q-subgroup of <g>Gp I Gp<g>, which implies that Q is normal in
<g>Gp. From this, we conclude that the p-complement H of <g> is
normal in <g>Gp. Since K s H and H is cyclic, it now follows
that K is normal in <g>Gp and so KGp IGpK. This, together
with PGp I GPP I GP, implies that P X K I <gn> permutes with GP,
and the proof is complete.

Remark: In view of Lemma 1.2.3, it is obvious that if
G I G under the isomorphism 9, then (an(G))e = ZGn(G).

The following example shows that every subgroup of an(G)
is not necessarily n-quasinormal in G. In particular, it shows that
if 63> and <b> are n-quasinormal in G, then <ab> is not
necessarily n-quasinormal in G.

Example: Let G I a,b><x>, where a3

ll
0‘
I!
....i
a)
O‘
l!

ax I a2, bx I b and x2 I 1. Clearly, ZGn(G) <a,b>. Hence
ab 6 ch(G). But ¢b> is not n-quasinormal in G since <ab>
does not permute with <x> which is a Sylow 2-subgroup of G. For,
otherwise <ab> would be normalized by x, an impossibility.

Let H be a subgroup of a group G. In general, there seems
to be no relationship between an(G) and ZGn(H)’ even if H is

normal in G. For example, if G IA and H is its Sylow 2-sub-

4
group, then an(G) I l$ an(H) I H. On the other hand, if G is

a nilpotent group and H is a proper (normal or non-normal) subgroup

28

I z a .
of G, then ZGn(H) H i Gu(G) G This leads us to investigate
the relationship between the generalized center of a group and the
generalized centers of its direct factors. We prove

Proposition 3.1.4: If G I H X K, then an(G) I
ZGn(H) x ch(K).

Proof: First we show that ZGn(H) X ch(K) s ch(G). Let
h be an element of H such that <h> is n-quasinormal in H and
G be a Sylow p-subgroup of G. Since G I H X K for some H

P P P P P
and Kp and since <h> is centralized by K, it follows that
<h>Gp I Gp<h>. This means that 4:) is n-quasinormal in G and
hence h E ZGn(G). Thus ZGn(H) s an(G)° Similarly, ZGn(K) s
an(G) and so an(H) X ch(K) is contained in ZGn(G).

Next to show that an(G) s ZGn(H) X ch(K), let g be an
element of G 'such that <g> is n-quasinonnal in G. Let P be
the Sylow p-subgroup of <g>. Clearly, P I dlk> for some p-elements
h 6 H and k 6 K. We will show that <h> is n-quasinornal in H.
For this, let Hq be any Sylow q-subgroup of H, q I p and x be
an element of Hq. By Theorem 3.1.3, P is n-quasinormal in G.
Hence it follows from Lemma 3.1.1 that P is normal in PG for
every Sylow q-subgroup Gq of G, q I p. In particular, x-1(hk)x I
(hk)n I hflkn for some integer n. This implies that h-n(x-1hx) I
kn(x-llo<) -1 E H n K I 1. Hence xdhx I hn and so <h> permutes
with every Hq for primes q I p. Now let Hp be any Sylow p-sub-
group of H. Then there exists a Sylow p-subgroup Gp of G such
that H SG . But G IH XK and therefore G IH x (K nG).

P P P P P
Since P I <hk> is n-quasinormal in G, it follows that P 5 GP.

Hence bk 6 Hp X (K 0 GP) , which implies that h 6 Hp. Consequently,

29

<h>Hp I Hfi<h> I Hp. This means that <h>: is n-quasinormal in H
. . H K .

and so h E an(H) Similarly, k 6 ZGn(K) Thus P S.ch( ) X an( )
Since ><g> is the direct product of its Sylow subgroups, it follows
that <g>'s ZGn(H) X ZGn(K). Hence an(G) s ZGn(H) X an(K)° This
completes the proof.

Remark: Mukherjee [15] has proved the above result for the
quasicenters under the added hypothesis of (\H‘, \K‘) I 1.

If G is nilpotent, then ZGn(G) I G but the quasicenter

Q(G) is not necessarily G itself. For example, let G I <a,b,c><x>,

2 2 2
where a I b I c I x2 I l, ab I ba, be I cb, ac I ca, ax I ab,

bx I b, cx I c. It is shown by Mukherjee [15] that \Q(G)| I 8.

But \G‘ I 16 and so Q(G) $G I ZGn(G). Using this example, we
can construct a non-nilpotent group G1 such that (“61) $ an(Gl) .

Let G1 =.G X S3, where G is the group defined above and

S3 18 the symmetric group on 3 letters. Then an(Gl) I an(G) X

an(S3) I G X A Since Q(Gl) $Q(G) X Q(SB) I Q(G) XA3, it follows

3.
that Q(Gl) $ ZGn(G1).

Next we determine the structure of the generalized center of
a group. For this, we need the following lemma.

Lemma 3.1.5: Every Sylow subgroup of an(G) of a group G
is generated by generalized central elements of G.

Ppppf; Let p1,p2,..., and ph be the prime divisors of

the order of an(G)' For 1 s i s n, denote by S the subgroup

1
generated by all pi-elements of G that are the generalized central

elements of G. Clearly, each S is n-quasinormal in G and hence

i

is subnormal in G. Furthermore, since a generalized central p1-

element of G belongs to every Sylow pi-subgroup P of C, it

1

follows that S 5 (Pi. Therefore, S

1 is a pi-subgroup of G.

i

30

Let pj and pk be any two different prime divisors of the
order of an(G)' Then, by the preceding paragraph and Lemma 3.1.1,
it follows that Sj is normal in Sij. Hence Sj and SR
and so 8 S is a subgroup for all primes p1 and pk with pj I pk

permute

J k
and 1 s j,k s n. Therefore, SISZ"'Sn is a subgroup which is con-
tained in an(G)' Now, let g be any generalized central element
of G. Since .<g> is the direct product of its Sylow subgroups
each of which is cyclic and n-quasinormal in C, it follows that every
Sylow subgroup of <g>’ is contained in some 31' Hence n<g> s
S1SZ"’Sn’ which implies that an(G) s $182...Sn. Thus an(G) I
8182...S . From this, the assertion in the lemma follows immediately.

n

Theorem 3,1;p; an(G) of a group G is nilpotent.

Ppppf: The proof of this theorem is essentially embodied
in that of the preceding lemma. For, if P is any Sylow subgroup
of an(G), then P is subnormal (in fact n-quasinormal) in G and
so P is a subnormal Sylow subgroup of ZGn(G). Hence P is normal
in an(G)’ which implies that an(G) is nilpotent.

Rppgpk: Note that every Sylow subgroup P of an(G) is
characteristic in G. This follows from the fact that P is char-
acteristic in an(G) and an(G) is characteristic in G. Also
note that G is nilpotent if and only if G I an(G)'

In the next few results, we prove some additional properties
of the generalized center.

Proposition 3.1.7: If N is a minimal normal subgroup of
G, then N sCG(ZGn(G)), the centralizer of an(G) in G.

P_ro_of_: If N n an(G) I 1, then clearly N s CG(an(G))°

On the other hand, if N n zcn(c) i 1, then N = N n zcnm) by

31

the minimality of N. This means that N s an(G) and so N is
nilpotent. Hence every Sylow subgroup of N is normal in G. But
N is a minimal normal subgroup of G and so N must be a p-sub-
group for some prime p. Let P be the Sylow p-subgroup of an(G) .
Then N is a normal subgroup of P. Hence N n Z(P) I l, where
Z(P) is the center of P. Since P is normal (in fact character-
istic) in G, Z(P) 4 G. Therefore, N n Z(P) 4G and so N n Z(P) I N.
Hence N s Z(P). The desired result is now obvious since an(G) is
nilpotent.

Theorem 3.1.8: If the smallest prime divisor of \G‘ divides
\zcn(o)‘, then Z(G) # 1.

P_l_.'pp_f_: Let p1,p2,..., and pH be the prime divisors of the
order of G, where p1 < p2 <...< pm. By hypotheiss, p1 divides

‘ZGn(G)| . Denote by P a Sylow {Di-Subgroup of G for i I 1,2,... ,n

i
*
and by P1 the Sylow pl-subgroup of an(G)' Then G I P1P2"'Pn
* a:
and P1 I 1. Since P: is characteristic in G, it follows that P1

at
is a normal subgroup of P Hence P1 0 Z(Pl) I l.

1.
Let y be a non-identity element of P: O Z(Pl). Then y
centralizes P1 and we shall show that every Pi for i > 1 is
also centralized by y. Let g be any generalized central element of
G such that g 6 P: and 2 be an element of P1 for i > 1. Lemma
3.1.1 yields that <g> is normalized by every pi-element of G. In
particular, <g><z> I <z><g> is a subgroup and <g> 4 <g><z>. But
<g><z> is supersolvable and p i > p1 and so <z> 4 <g><z>. Thus
zg I gz. From this we see that y centralizes 2 since y belongs

9:
to P1 which is generated by generalized central elements of G.

Hence y E Z(G) and the theorem is proved.

32

Corollary 3.1.9: Let p be the smallest prime divisor of
\ch(G)‘. If every prime divisor q of \G‘ with qi< p does not
divide p-l, then Z(G) I l.

Ppppf; Let P be a Sylow p-subgroup of G and P* be the
Sylow p-subgroup of an(G)° As in the theorem, it follows that there
is an element y in P*.r]Z(P) such that y-I l and y centralizes
P and all Sylow r-subgroups of G for primes r >»p.

Let 2 be a q-element of G with prime qi< p and g be a
generalized central element of G such that g 6 P*. By Lemma 3.1.1,
<g> is normalized by z and so <g><z> is a subgroup. Since (2)
is a Sylow q-subgroup of <g><z>. and q does not divide p-l, it
follows from the Sylow theorems that 1(2) is normal in <g><z>.

Hence gz I zg, which implies, as in the theorem, that y centralizes
z and so y E Z(G).

Theorem:gpl.10: If p is a prime divisor of \ch(G)\ and
g is an element of an(G) such that all the prime divisors of
\g‘ are less than p, then g centralizes every p-element of G.

gpppf; we may assume that g is a q-element and prime
q < p. we may also assume that g is a generalized central element
of G since the Sylow q-subgroup of an(G) is generated by such
elements. Let x be a p-element of G. By Lemma 3.1.1, <g><x> is
a subgroup and <g> 4 <g><x>. Since <g><x> is supersolvable and

p > q, <x> 4 <g><x>. Thus get I xg.

3.2 Generalized Hypercenter
In this section we generalize the concept of the hypercenter
and hyperquasicenter to that of the generalized hypercenter and study

the structure and some properties of this new hypercenter.

33

Defipition 3.2.1: For a group G, let (ZGn(G))0 I 1 and
(an(G))i+l/(an(G))i be the generalized center of GI(ZGn(G))1.

Then we define the generalized hypercenter Z;n(G) of G to be the
terminal member of the ascending chain of characteristic subgroups:
1 = <zcn<c>>o < an<6> = (2611(6))1 < (ZGn(G))2 <...< swam»m =
zgnm).

It is obvious that 2;;(6) contains the hypercenter and
hyperquasicenter of G. The hypercenter 2*(6) is always nilpotent
but 2;;(0) is not necessarily nilpotent. The generalized hypercenter
2;;(33) of the symmetric group 83 on three letters is 83 itself
which is not nilpotent. However, we shall show that 2:;(G) is super-
solvable.

The following example shows that the hyperquasicenter of'a
group can be trivial even though the generalized hypercenter is non-
trivial.

Example: Let G be the symmetric group S wreathed by the

3
cyclic group C3 of order 3, i.e., G is the direct product of three
copies of 83 extended by an automorphism of order 3 which permutes
the copies in a cycle. For this group, Q(G) I Q*(G) I l but an(G) a
z;g(c) has order 27. I

The hypercenter Z*(G) of a group G is known to be the
intersection of all normal subgroups N of G such that Z(GIN) I I.
A similar result is true for the generalized hypercenter.

T_h£orem 3.2.2: For a group G, 2311(6) I an‘N 4G and
anm/N) =- i).

Ppppf; Consider the chain: 1 I (ZGn(G))O‘< an(G) I

(ch(G))1 < (an((:))2 <...< (2mm)m - zznm) and let

34

- 9:
T = n{N\N 4 c and anm/N) = 1}. Then T s zcnm) since

9: - *
an(G/an(G)) - 1. To prove ZGn(G) s T, let K be a normal sub—
group of G such that an(G/K) I I and g be a generalized
central element of G. From Lemua 1.2.3, it follows that gK is a
generalized central element of GIK. Hence gK is an element of
an(G/K) I I and so g belongs to K. This implies that ZGn(G) I
(ZGanI s K.

Next we assume that (an(G))i s K, 1 s i.< m, and wish to
show that (an(G))i+l s K. Let x(ZGn(G))1 be a generalized central
element of G/(ZGn(G))i' Since GIK. is a homomorphic image of
G/(an(G))i’ it follows that xK is a generalized central element of
G/K. But ZGn(G/K) I l and so x belongs to K. This means that
an(c/(zcn(c))) = (2611(0)) ll.|_1/(zca.m(c))1 s K/(an(c))i). Hence

*
(sz(G))1+1 s K, which implies that ZGn(G) s K. From this, it
*

follows that an(G) s T and the proof is complete.

The next proposition follows from the above theorem.

Proposition 3.2.3: Let N be a norml subgroup of G. Then,

* *

an(c)N/N s an(G/N).

Proof: Let KIN be a normal subgroup of GIN such that
ZGn(G/NIK/N) = i. Since GIN/KIN I G/K, it follows that an(G/K) I i.

'k '1:

Hence zcnm) s K and so an(c)N/‘N s KNIN =- KIN. This means that
4: .. a:
ZGn(G)N/N s n{K/N\K/N <1 GIN and ZGn(GIN/KIN) = 1}... ZGn(GIN).

Remark: The inclusion in the above proposition could be
proper, too. For example, consider G I A4, the alternating group
of degree 4, and let N be the Sylow 2-subgroup of A . Then

4
* C *
an(c)N/N = 1 $ anm/N) = GIN.

35

The following result which we shall need later is a special
case of the preceding proposition.
Preposition 3.2.4: Let T be a normal subgroup of G such
h 2* Th 2* / 2* I
t at T s Gn(G). en Gn(G T) I Gn(G) T.
Proof: Consider the chain: '1' = T/T < w1/T < 91le <...< WmIT =
*
an(G/T), where W1/T - an(G/T) and Wi/T/W1_1IT = an(G/T/wi_1/T)
*
for i I 2,3,...,m. Since T s an(G)’ it follows from Proposition
* *
3.2.3 that ZGn(G) s Wh. To prove that Wh sLZGn(G), we first show
*
that W1 5 an(G)' For this, let gT be a generalized central

*
element of GIT. Then gZGn(G) is a generalized central element of

* *
GIZGn(G) since GIZGn(G) is a homomorphic image of GIT. Hence g

* *
belongs to ZGn(G), which implies that w1 s ZGn(G).
*
We now assume that w1 s.ZGn(G), 2 s i‘<‘m, and will show
*
that W1+1 s ZGn(G). Let 9 be the natural isomorphism from

GIT/wi/T onto G/Wi. It follows that an(G/w1) = (an(GITIw1IT))e I

IWi. Now replacing T by W in the argument used above, one

1
5 2* G H 'W 2*
Gn( )- ence m 5 G“(6) and so wIn

Wi+1
can show that W“1
2* h h ha 2* (G IT - 2* (GIT

Gn(G), w ic means t t Gn ) - Gn ).

In general, there is no relationship between the generalized
hypercenter of the group and the generalized hypercenter of a sub-
group of the group. This is confirmed by the following examples.

(I) Let H be a proper subgroup of the symmetric group

* x *
S3 on three letters. Since an(S3) I S3, we have ZGn(H) $LEGn(S3).

* * *
(2) Let G I A Since an(G) - l, an(G) $ an(H) for

4.
every proper subgroup H of G.

The last example suggests the next result.

36

Proposition 3.2.5: Let H be a subgroup of a group G such

* * *
that ZGn(G) s H. Then an(G) £.ZGn(H). In particular,
2* 2* 2* G
Gn( Gn(G)) = Gn( )'

*

Proof: We use induction on the order of G. Since ZGn(G) S‘H,
ZGn(G) s H. This means that all generalized central elements of G
are in H. Now it follows from Theorem 1.2.5(i) that all generalized
central elements of G are also the generalized central elements of

*
H. Hence an(G) s ch(H) s ZGn(H) and so, by Proposition 3.2.4,
2* H/Z 2* H /z 2* / 2* (G lz G
Gn( Gn<G>> . Gn( ) Gn(G)' Since Gn(G an(G)) - Gn ) Gn( ) s
* *
Z Z I
HIZGn(G), it follows by induction that an(G)/zcn(c) s Gm(li/ Gn(G))
* * *
an(u)/zcn(c), which implies that an(G) 5 anal).
*

To prove the particular case, set H I ch(G). This yields
h 2* G) Z* 2* 2*
t at Cu( 5 Gn( Gn(G)) s Gn(G).

It is easily proved that if G and G. are isomorphic under

* -— *
the map 9, then an(G) is the image of an(G) under 9. Using
this and Proposition 3.1.4, we prove the following result.
* *
Proppsition 3.2.6: If G I H X K, then Z (G) I 2 (H) X
Gn Gn

2* K
Gn( )'

Proof: We use induction on the order of G. Clearly, we may
assume that ZGn(G) I an(H) X an(K) I 1. By induction and Proposi-
* * *
tion 3.2.4, an(H/ch(H) x Klzcnao) - anm/zcnmn x an(K/zcn(x)) ..

* *
an(H)/an(H) x ZGn(K)IZGn(K). Let 9 be the natural isomorphism
*
from G/an(G) onto HIZGn(H) x x/zcnao. Then, zcn(u)/zcn(u) x
2* KIZ K 2* GM G 9 * /z 9 '1
Gn( ) Gn< ) = ( Gn( Gn( ))) .. (Zena?) Gm(6)) - Now apply 9
* * *
to obtain an(G)/an(G) = (2mm) x an(K))/zcn(c), which yields

the desired result.

37

Remark: If G is an extension of H by K, then it is not
true in general that 2:n(G) I z:n(H)-Z;n(K), as confirmed by A4.
We now investigate the structure of the generalized hyper-
center. Our aim here is to prove that it is supersolvable. For
this, we need the following theorem. However, we first recall
a definition from [8].

Definition 3.2.7: Let p1 > p2 >...> pn be the natural
ordering of the prime divisors of the order of a group G and P1
be a Sylow pi-subgroup of G for i I 1,2,...,n. Then G has the
Sylow tower property of supersolvable groups if for each k,

l s k s n, P1P2°"Pk is a normal subgroup of G.

Let us point out that a group having this property is
necessarily solvable.

Theorem 3.2.8: Z;n(G) of a group G has the Sylow tower
property of supersolvable groups.

gpppf; we proceed by induction on the order of G. Thus,
in view of Proposition 3.2.5, we may assume that z;g(c) I G. Let
p be the largest prime divisor of the order of G and Gp be a
Sylow p-subgroup of G. We shall show that Gp is normal in G.

First suppose that p does not divide the order of G/ZGn(G).
Then, Gp S.ch(G). Since an(G) is nilpotent, it follows that G1)
is normal in G. Next suppose that p divides the order of G/an(G)'
Since p is the largest prime divisor of the order of GIZGn(G) and
z:n(G/ZGD(G)) = z;n(G)/zcn(c) = G/an(G) , by induction (G/an(G))p =
szGn(G)/zcn(c) is normal in G/an(G). This means that szGn(G) 4 Q,
But the Sylow subgroups of an(G) are normal in G. Hence

a - Z .
GPZGn(G) GEL <IG, where L is the p complement of Gn(G) To

38

prove that G1) 4 C, it suffices to show that L s NG(GP) since

G I GPLNG(GP) I LNG(Gp) by Frattini Lemon. For this, let x be
any q-element of G for a prime q I p such that <x> is fi-quasi-
normal in G. By Lemma 3.1.1, <x> d <x> GP. Let g be any element
of GP. Then <x><g> is a supersolvable subgroup of G and

ct> 4 oc><g>. Since p > q, <g> is also normal in <x><g>. Hence
g and x centralize each other and so x E NG(GP) . But, by Lemma
3.1.5, all such x generate L. Hence L g NG(Gp) , which implies
that G1) is normal in C.

To complete the argument, consider GIGp. By induction,
Z;n(GIGp) I G/Gp has the Sylow tower preperty of supersolvable
groups, which clearly implies that G I z;n(G) has the same property.
This proves the theorem.

We are now ready to prove the supersolvability of z;n(G) .

Theorem 3.2.9: Z;n(G) of a group G is supersolvable.

Pgo_of_: We proceed by induction on the order of G. If
z;n(G) $ G, then z;n(z;n(c)) I 2;n(G) is supersolvable by induction.
Hence we may assume that z;n(c) I G. This means that every factor
group GIK of G for K I l is supersolvable by induction since
z;n(GIK) = z:n(G)/K = GIK.

If the Frattini subgroup Q(G) of G is not identity, then
GIMG) is supersolvable. Now a theorem of Huppert [9] yields that
G is supersolvable. So we assume that Q(G) I 1. Let p be the
largest prime divisor of |G| . By Theorem 3.2.8, the Sylow p-sub-
group P of G is normal in G. Hence Q(P) s Q(G), which implies
that Q(P) I 1. This means that P is elementary abelian. Let M

be a maximal subgroup of G. We will show that [G:M]‘ is a prime.

39

It follows from Theorem 3.2.8 that G is solvable. Hence
[G:M] is a power of a prime. If [G:M] is not a power of prime p,
then P s M and, since GIP is supersolvable, [GIP:MIP] I [G:M]
is a prime. On the other hand, if [G:M] is a power of p, we con-
sider ZGn(G) and proceed as follows: It is obvious that
ZGn(G) I 1. If a prime q I p divides the order of an(G)’ then
the Sylow q-subgroup Q. of an(G) is normal in G and therefore
6.5 M. But 6/6. is supersolvable, which means that [G/QTMIQ] I
[G:M] is a prime. Hence we may assume that an(G) is a p-subgroup.
Since an(G) is generated by the generalized central elements of
G and the powers of a generalized central element of G are again
the generalized central elements of C, it follows that G contains
generalized central elements of order p. Let N be the subgroup
generated by all these elements of order p. Since the conjugates
of a n-quasinormal subgroup are n-quasinormal in the group, it follows
that N is normal in G. If N s M; then, as before, [G:M] is a
prime. On the other hand, if N £ M, then there is an element y of
order p such that <y>) is n-quasinormal in G and y i M. Since
P is abelian, <y>’ is a normal subgroup of P. It follows from
Lemma 3.1.1 that <y>» is normalized by p'-e1ements of G. Hence
<y> is normal in G. This means that ‘M<y>' is a subgroup and so
M<y> I G, which implies that [G:M] I |q>‘ I p, a prime. Now by a
theorem of Huppert (Theorem 1.1.13), G I 2;;(6) is supersolvable
and the proof is complete.

The above result leads to the following simple observation.

Theorem 3.2%195 .A group> G is supersolvable if and only if

*
G I an(G)°

4O

*
Proof: If G I an(G)’ then G is supersolvable by the pre-
ceding theorem. On the other hand, if G is supersolvable, then G
has a cyclic normal subgroup <g>'I 1. Since a normal subgroup is

*
always n-quasinormal, it follows that <g> s ZGn(G) s ZGn(G). Hence

* /
an (G <g>)

2* G
so an( )

* *-
an(G)/<g>. But by induction, an(G/<g>) = GI<g> and

G.

In the next result, we obtain a condition for a group to be
supersolvable.

Theorem 3.2.11: A group G is supersolvable if and only if
GIZE£(G) is supersolvable.

2522f: If G is supersolvable, then its factor group
GIZ;n(G) is supersolvable, too. Conversely, if G/Z;g(G) is super-
solvable, then by Theorem 3.2.10, Z:;(GIZEQ(G)) I GIZS;(G). But
z;n(G/z;g(s)) is identity and therefore G I 228(G). Since Z;g(G)
is supersolvable, it follows that G is supersolvable.

It is known that the product of two supersolvable subgroups
is not necessarily supersolvable. However, if one of the subgroups
is the generalized hypercenter, then the following is true.

Theorem 3.2.12: Let S be a supersolvable subgroup of a
group G. Then 82;;(G) is supersolvable.

meg: We use induction on \G| . If $2;n(G) $G, then
SZ:n(SZ:P(G)) is supersolvable by induction. Proposition 3.2.5
yields that zznm) s z;n(sz;n(c)) , which implies that 323nm) is
supersolvable. So assume that 82:Q(G) I G. Since GIZ;;(G) I .
sz* ((2)/2* (c) '5 s/s n 2* (G) and 2* (GIZ* (G)) is identity, it

On Gn Gn Gn Gn
follows that Z:;(SIS (12;;(G)) is identity. But SIS FIZ;;(G)

* * *
is supersolvable and so an(8/s (\ZCD(G)) I SIS Flzcn(G). This

41

means that 5/8 FIZ;n(G) is identity, which implies that S s 2;;(c).
Hence 82;;(G) = z;n(c). But Z;£(G) is supersolvable and there-
fore SZE§(G) is supersolvable.

Baer proved in [4] that the hypercenter of a group G is
the intersection of the maximal nilpotent subgroups of G. Since the
generalized hypercenter is supersolvable, it seems reasonable to con-
jecture that the generalized hypercenter of G is the intersection
of the maximal supersolvable subgroups of G. we have not yet been
able to prove this conjecture completely. However, we prove part of
the conjecture in the following theorem.

Theorem 3.2.13: 2;;(G) of a group G is contained in the
intersection of the maximal supersolvable subgroups of G.

2322;; we must show that z;g(c) is contained in every
maximal supersolvable subgroup of G. For this, let M. be any
maximal supersolvable subgroup of G. By Theorem 3.2.12, M2;#(G)
is supersolvable. Hence either ME;;(G) I=G or M2;n(G) I'M. If
M2;n(G) I G, then G is supersolvable. This implies that M -=G
and therefore Z;n(G) s M. On the other hand, if MB;n(G) I'M, then
it follows immediately that z;n(c) s M.

ggmaggg Note that many of Mukherjee's results (see [15] and
the introduction of this chapter) on Q(G) and Q*(G) are immediate
consequences of our results on an(G) and ZZ£(G), since Q(G) s
an(G) and Q*(G) s z;n(c) and since the conclusions have already
been proved for an(G) and z;#(G).

we conclude this section with a result which may be of some
independent interest. It gives a condition for a group to have the

Sylow tower property of supersolvable groups.

42

Theorem 3.2.14: A group G has the Sylow tower property of
supersolvable groups if and only if G/Z;s(G) has the same property.

2322;; We need only prove that if G/Z:n(G) has the Sylow
tower property of supersolvable groups, then G has the same property.
For this, we first show that if for any group H, H/ch(H) has the
Sylow tower property of supersolvable groups, than H has this pro-
perty, too. We proceed by induction on the order of H.

Let N be a normal subgroup of H. we shall show that
H/N/ZGn(H/N) has the Sylow tower property of supersolvable groups.
Since H/N/ZGn(H)N/N a H/ZGn(H)N and H/ch(H)N is a homomorphic
image of H/an(H)’ it follows that HIN/ZGn(H)N/N has the Sylow
tower prOperty of supersolvable groups. Hence HIN/ZGnUI/N) has the
same prOperty because H/N/ZGn(H/N) is a homomorphic image of
HIN/ZGn(H)N/N by Proposition 3.2.3.

Let p be the largest prime divisor of the order of H and
Hp be a Sylow p-subgroup of H. One can easily verify by essentially
duplicating the argument used in Theorem 3.2.8 that Hp is a normal
subgroup of H. Now it follows from the observation made in the pre-
ceding paragraph that H/Hp/ZGn(H/Hp) has the Sylow tower property
of supersolvable groups. Hence by induction, H/Hp has this property
which, in fact, implies that H has this property.

To establish the theorem, consider the chain: 1 I (Z‘Gn(G))o <
zcnm) = (zcmam1 < (an(c))2 <...< (20:36))In - zznm), where
(an(c))i/(zcn(c))i_1 = an(c/(an(c))1_1), 1 s i s.m. Since
6/2346) = cl<an<c)>m = c/(zcnmn Iczcnm))mn/(zwc))m_1 =

c/(zcn(c))m_1/an(c/(zcn(c))m_1) and c/z;g(c) has the Sylow tower

m-l

property of supersolvable groups, it follows by what we have shown

43

above that G/(an(G))m-l has the Sylow tower prOperty of super-
solvable groups. Now repeated use of this argument shows that G

has the same property, the desired conclusion.

3.3 Generalized Hypercentral Subgroups

In this section, the generalized hypercenter is shown to be
the product of all generalized hypercentral subgroups. The notion
of the generalized hypercentral subgroups is an extension of the
concept of the hypercentral subgroups introduced by Baer. He proved
that the hypercenter is the product of all normal hypercentral sub~
groups. In our case the normality is included in the definition.

Definition 3.3.1: We shall call a normal subgroup H of a

 

group G a generalized hypercentral (GHécentral) subgroup of G
if for all M $ H and M normal in G, H/M n an(G/M) i i.
Proposition 3.3.2: If H is a GH-central subgroup of G,
then for each N 4 G and N $ H, H/N is a GH-central subgroup of
GIN.
M: Let K/N $H/N and K/N ans/N. Then K$H and
K 4 G. Hence H/K n ZGn(G/K) I 1. Consequently, there is an element
hK of H/K such that hK e an(s/x) and h 4 K. suppose G/K 5'
GIN/KIN under the map 9. Since ZGn(G/N/K/N) a (zcn(c/K))°, it
follows that (hK)e is a non-identity element of HIN/KIN n
ZGn(G/N/KflN), which implies that HIN is a GH-central subgroup of GIN.
Leuma 3.3.3: If H s an(G) and H 46, then H is a GH-
central subgroup of G. In particular, an(G) is a GH-central sub-

group of G.

(.4

Proof: Suppose K 4 G and K $ H. Since G/K is a homo-
morphic image of G, it follows from Lemma 1.2.3 that an(G)KIK I
an(G)/K s zcn(o/1<). But H/K s an(c)/1<. Hence H/K n ZGn(G/K) =
H/K ,4 i.

Theorem 3.3.4: For 1 s i s m, every member (an(G))i of

 

the chain: 1 I (an(G))O < ZGn(G) I (an(G))1 < (an(G))2 <...<
(ZGn(G))m = zgnm) is a GH-central subgroup of s.

Proof: We use induction on \G| . In view of the preceding
Lemma, we prove the theorem for i I 2,3,...,m. Consider the group
G/ZGn(G) and form the chain: i = (an(c))1/zcn(c) < (an(c))2/zcn(c) <
...< (an(c))m/zcn(o) = z;n(G)/zcn(c), where an(c/an(c)/(zcn(c))14/
an(c)) = (an(c))i/ZGnm)/(an(c))1_1/zcn(s) for i - 2,3,...,m.
Now suppose M $ (ZGn(G))i and M 4 G, i > 1. If ZGn(G) s M, then
(ZGn(G))1/zcn(c) IM/ZGn(G) n an(c/zcn(c)/M/zcn(c)) i i since by
induction, (ZGn(G))i/ch(G) is a GH-central subgroup of G/ZGn(G).
From this we see that (an(G))i/M n an(G/M) ,4 1. On the other hand,
if ZGn(G) t M, then there is a generalized central element g of
G such that g f M. By Lemma 1.2.3, gM E an(G/M) and so
gM 6 (an(G))1/M n ZGn(G/M) . Hence (an(G))i is a GH-central sub-
group of G.

Lemun 3.3.5: If N1 and N2 are GH-central subgroups of a
group G, then the product NlN2 is also a GH-central subgroup of G.

1593;: Let M<G with M$N1N2. If M$N1 or M$N2,
then N1N2/M n an(G/M) 5‘ 1 since NilM n ZGn(G/M) :‘ l for i I 1
or 2. If M an IN1 and M 0N2 INZ, then NlN2 sM, which is
impossible. Thus we may assume without loss of generality that

M n N1 = w $N1. Since w is also normal in c, Nllw n anmIW) .4 1.

45

Let xW be a non-identity element of N1/W'F\ZGn(G/W). Since G/M
is a homomorphic image of G/W and x 1 M5 xM. is a non-identity
element of NiN2/M n ZGn(G/M). Hence Nl'N2 is a GH-central subgroup
of G.

From the above results we now obtain the following characteriza-
tion of the generalized hypercenter.

Theorem 3.3.6: 2:6(G) of a group G is the product of all
GH-central subgroups of G.

M: It follows from Lemma 3.3.5 that the product P of
all GH-central subgroups of G is GH-central. Hence z;g(c) siP
by Theorem 3.3.4. If 2: (G) $P, then P/z* (G) n z (c/z* (G)) r '1'

n Gn Gn Gn

which means that ZGn(G/Z;g(G)) * I, an impossibility. Thus

2* (G) - P
Gn °

CHAPTER IV

FINITE GROUPS WHOSE i—th MAXIMAL
SUBGROUPS ARE n-QUASINORMAL

A number of mathematicians have studied the structure of the
group when its i-th maximal subgroups satisfy some imbedding prOperty.
The most natural imbedding property is perhaps normality. The other
examples of this prOperty are quasinormality, n-quasinormality and
subnormality. Huppert [9] and Janko [12] have described the structure
of the group when its i-th maximal subgroups are normal and Mann [14]
has studied the structure of the group when its i—th maximal subgroups
are subnormal or quasinormal. In this chapter, we investigate the
structure of the group when its i-th maximal subgroups are "Iquasi-
normal. Our primary concern is to improve the following theorems.

(Huppert [9]). If each second maximal subgroup of G is
normal in G, then G is supersolvable. If the order of G is
divisible by at least three different primes, then G is nilpotent.

(Huppert [9]). Let each third maximal subgroup of G be
normal in G. Then: (i) G' is nilpotent; (ii) the rank of
G I r(G) s 2; (iii) if ‘6‘ is divisible by at least three different
primes, then G is supersolvable.

(Janko [12]). Let G be solvable. If each fourth maximal
subgroup of G is normal in G, then: (i) r(G) s 3; (ii) if ‘G‘
is divisible by at least four distinct primes, then G is super—

solvable.
46

47

(Mann [14]). Let G be solvable, and each n-th maXimal sub-
group of G be quasinormal in G. Then: (i) r(G) s n-l; (ii) if
‘G‘ is divisible by at least n-k+1 distinct primes, then r(G) s k,
where k 2 1.

In Section 2, we prove the above results under the weaker
assumption that each i-th maximal subgroup (i I 2,3,4) be n-quasi-

normal instead of normal or quasinormal.

4.1 Definitions and Assumed Results

Definition 4.1.1: For a group G, a series of subgroups

G =G >G1>G2 >...>G =1

0 k

is called a chief series if every G is a maximal normal subgroup

i
of G contained in Gi-l for i I 1,2,...,k. The factor groups
Gi/Gi+1 are called the chief factors of G.

It is known that if G is solvable, then every chief factor
of G is an elementary abelian p-group for some prime p.

Definition 4.1.2: Let G be a solvable group. Then the
rank of G, denoted by r(G), is the maximal integer n such that
G has a chief factor of order pn for some prime p.

We list for an easy reference two known results of which we
shall make rather frequent use.

54.1.32 Huppert [8]. If all proper subgroups of the non-
nilpotent group G are nilpotent, then G is solvable; ‘G‘ I p'aqb
for distinct primes p and q; the Sylow p-subgroup Gp is normal

and each Sylow q-subgroup Gq is cyclic.

48

(4.1.4) Doerk [6]. If each maximal subgroup of G is super-
solvable, then: (i) G is solvable; (ii) G has a Sylow tower for
the natural (descending) ordering of prime divisors of ‘G‘, or G
satisfies the hypotheses of (4.1.3); (iii) if G itself is not

supersolvable, then G has exactly one normal Sylow subgroup.

4.2 Generaligengesults

For a group G, we prove the following theorems:

Theorem 4.2.1: If each second maximal subgroup of G is
n-quasinormal in G, then G is supersolvable. Furthermore, if
‘G‘ is divisible by at least three different primes, then G is
nilpotent.

Theorem 4.2.2: If each third maximal subgroup of G is
fl-quasinormal in G, then:

(i) if ‘Gl is divisible by three or more different primes,
then G is supersolvable;

(ii) the commutator subgroup G' of G is nilpotent;

(iii) the rank of G I r(G) s 2.

Theorem 4.2.3: Let G be solvable. If every fourth maximal
subgroup of G is n-quasinormal in G, then:

(i) if \G\ is divisible by four or more different primes,
then G is supersolvable;

(ii) r(G) s 3.

Proof of theorem 4.2.1: Let M. be a maximal subgroup of G.
Then every maximal subgroup of M is n-quasinormal in G. This means
that all maximal subgroups of MI are n-quasinormal in M. by Theorem

1.2.5(i) and therefore they are normal in M by Theorem 1.2.5(ii).

49

Hence M is nilpotent and so all proper subgroups of G are nil-
potent. Now by (4.1.3), G is solvable. In addition, if \c\ is
divisible by three or more different primes, then G is nilpotent
and we have diSposed of this case.

Next we consider the case where ‘G‘ is divisible by, at
most, two distinct primes. To prove that G is supersolvable, we
must show that [GzM], the index of M. in G, is a prime for an
arbitrary but fixed maximal subgroup M. of G since a theorem of
Huppert states that a group is supersolvable if and only if its
maximal subgroups have prime index. If Mb I 1, then, since by ‘
Lemma 1.2.3 G/Mb satisfies the hypothesis of the theorem, G/MG
is supersolvable by induction. From this it follows that
[G/Mb:M/Mb] I [G:M] is a prime. Therefore, we may assume that

M I 1, and form the nnximal chain: M < M < G, where M is

G l l
maximal in M. Since M1 is n-quasinormal in G, by Theorem 1.2.5
(iii) M

1 is subnormal in G. Hence ‘M1 s MSG I M.G I 1, which
implies that M1 I 1. But M is nilpotent and, therefore, \M‘ I p,
a prime. Now consider [GzM] which is a power of a prime since

G is solvable. If [G:M] is a power of p, then G is a p-group
and we are finished.‘ On the other hand, if [G:M] I qm, q I p, then
\G‘ I pdm. Let Gq be a Sylow q-subgroup of G and L be a
maximal subgroup of Gq. Then Gq is maximal in G, and L is
n-quasinormal in G. Since 'M is a Sylow p-subgroup of G,

LM I ML is a subgroup of G. But M is maximl in G and LM !‘ G.
Therefore, LM I M. This implies that L s'M .and so L I 1. Hence
\Gq| I q showing that LG:M] I q, a prime. This completes the

proof.

50

Eggggk: If we simply require that every second maximal
subgroup of G be subnormal in G, then G is not necessarily
supersolvable, as confirmed by A4, the alternating group of degree 4.

Proof of Theorem 4.2.2: (1) From.Theorem.l.2.5(i) and
Theorem 4.2.1, it follows that every maximal subgroup of G is
supersolvable. Hence G is solvable by (4.1.4). Moreover, if the
order of G is divisible by at least four different primes, then
G is supersolvable. Thus we need only consider the case in which
\G‘ is divisible by three different primes. Before proceeding, it
should be noted that every second maximal subgroup of G is nil-
potent by Theorem l.2.5(i) and (ii) and, therefore, every third
maximal subgroup of G is also nilpotent.

Let \G‘ I panrY, where p >.q >tr and a,8,y >10.

Suppose that G is not supersolvable. Then, since (4.1.3) does not
hold, it follows from (4.1.4) that the Sylow p-subgroup G1) is

normal in G and no other Sylow subgroup of G is normal in G.
Since G is solvable, there exist Sylow subgroups Gq and Gr

such that GqGr is a subgroup. Let H I GqGr' If H is not maximal
in G, then Gq is contained in a third maximal subgroup of G.

Since each third maximal subgroup is nilpotent and subnormal (being
n-quasinormal; Theorem l.2.5(iii)), it follows that Gq is sub-
normal in G. But a subnormal Sylow subgroup is always normal and

so Gq is normal in G, a contradiction. Hence H is maximal in G.

Now suppose that a 2 2. Since every maximal subgroup of G
is supersolvable, H is supersolvable, too. Therefore, Gr is pro-
perly contained in a maximal subgroup of H. This means that Gr

is contained in a third maximal subgroup of G. Hence, as before,

51

Gr is normal in G, again a contradiction. Thus 9 I 1. By a
similar argument, y I 1 and so ‘Gi I paqr. Next suppose that L
is a maximal subgroup of Gp and consider the following maximal

chain:
L‘< G '< G G <:G
P P q

From this we see that L is n quasinormal in G. Hence L permutes
with H and therefore LH is a subgroup. Since H is maximal in

G and LH 9‘ G, LH I H. Thus L s H and so L I l, which means that
a I 1. Hence G is supersolvable, a contradiction to our assumption
that G is not supersolvable. Therefore, we have the desired result.

(ii) In view of part (i) and the fact that the commutator

subgroup of a supersolvable group is always nilpotent, we may assume
that G is not supersolvable and \Gl is divisible by two different
primes p and q. we may further assume without loss of generality
(see (4.1.4)) that Gp is normal in G. Then Gq is not normal in
G and we will show that Gq is either abelian or cyclic.

Suppose (sq)G 9‘ 1. Since \G/(Gq) is divisible by both

Gl
primes p and q, (G/(Gq)G)' is nilpotent by induction. Clearly,
(G/Gp)‘ is nilpotent. Since (clap)' I c'lc' nop and (G/(Gq)G)' I
G'IG' n (Gq)G, it follows that G'/(G' n GP) n (G' n (Gq)G) I G'

is nilpotent. Next suppose that (Gq)G I 1. If Gq is maximal in

G, then every second maximal subgroup of Gq is n-quasinormal (hence
subnormal) in G. Since (Gq)G I (Gq)SG I 1, all second maximal sub-
groups of Gq are 1. Therefore, \Gq‘ S q2 ‘which implies that Gq

is abelian. On the other hand, if Gq is not maximal in G, then

there exists a maximal subgroup M of G such that Gq < M < G.

52

Now if Gq is not maximal in M, then, as in part (i), Gq is normal
in G, a contradiction. Therefore, an< Mh< G is a maximal chain.
Hence every maximal subgroup of Cd is subnormal (being n-quasi-
normal) in G. Since Gq is not subnormal in G, Gq must have a
unique maximal subgroup and so Gq is cyclic. Now to show that

G' is nilpotent we need only note that G' sLGp since G/Gp (3 Gq)
is abelian. This proves part (ii).

(iii) Again, the only case that requires a proof is the one
in which \G‘ is divisible by two distinct primes p and q. we
further assume that G is not supersolvable, otherwise r(G) I 1.
As in part (ii), we suppose that Gp is the only Sylow subgroup of
G which is normal in G.

Let Ni be normal in G and N1 I 1 for i I l and 2.
If p and q both divide ‘G/Ni" then by induction r(G/Ni) s 2,

and if G/N1 is a p or q-group, then r(G/Ni) I 1 s 2. Hence if

N1 F\N2 I 1, then r(GlN1 FINZ) I r(G) max{r(G/Ni)} s 2 and we are
done. Thus we may assume that G has a unique minimal normal sub-
group N. Since Gp is normal in G, N is a p-subgroup. It now
suffices to show that \N‘ s p2 because we already have r(G/N) s 2
by induction.

Let Gq be a Sylow q-subgroup of G. If N I GP, then
NGq I G.‘ Hence NGq is supersolvable. Since Gp is normal in G,
it follows that its center Z(Gp) is normal in G. Thus ‘N s Z(Gp)
and so every subgroup of N is normal in GP. Let T (f 1) be a
normal subgroup of NGq such that T s N. Then T is normal in

GP. Hence T is normal in Gqu I G and so T I=N by the unique-

ness of N. Therefore, N is also a minimal normal subgroup of NGq,

53

which implies that \N‘ I p < p2. On the other hand, if N I GP,
then Gp is abelian and Gq is maximal in G. Since G has a
unique minimal normal subgroup, it follows that (Gq)G I (Gq)SG I 1.
Hence every second maximal subgroup of Gq is 1 and so |Gq‘ s qz.
First, suppose that \Gq‘ I q2 and consider the following maximal

chain:

L < C G K. G
P<P< a

where L is maximal in Gp and K is maximal in Gq. Now* L is
n-quasinormal in G, and so LGq is a subgroup of G. But LGq I G,
and therefore LGq I Gq. From this we conclude that L I 1, which
shows that \N\ I ‘GP‘ I p s p2. ‘Next, suppose that ‘Gq‘ I q. Then,
in the same manner, it follows that every second maximal subgroup of
Gp is l, which haturn proves that \N‘ I ‘GP‘ s p2. This completes
the proof of part (iii) and of the theorem.

ggmggh: The group ‘A4 shows that if the order of a group
is divisible by two different primes and if its third maximal sub-
groups are n-quasinormal, then the group need not be supersolvable
in general.

Proof of Theorem 4.2.3: (1) Let M. be an arbitrary but
fixed maximal subgroup of G. By Theorem 1.2.5(i), third maximal sub-
groups of M. are n-quasinormal in M. Since \G‘ is divisible by
at least four different primes and G is solvable, {M\ is divisible
by at least three different primes. Hence by part (i) of Theorem
4.2.2, M is supersolvable. Now G is supersolvable by a theorem

of Huppert [9].

54

(ii) We use induction on the order of G. In view of part
(i), the only cases that need proof are the ones in which \G‘ is
divisible by three and two different primes, reapectively. we treat
these cases separately. Before proceeding, we should observe that
each second maximal subgroup of G is supersolvable by Theorem
1.2.5(i) and Theorem 4.2.1. Also note that every third maximal sub-
group of G is nilpotent. This follows from Theorem 1.2.5(i) and
(ii) and the fact that a group is nilpotent if and only if its maximal
subgroups are normal.

Case 1. \G‘ is divisible by two primes, p and q. Then,
as in part (iii) of Theorem 4.2.2, we can assume that G has a unique
minimal normal subgroup N and by induction r(GfN) s 3. Without
loss of generality, let \N\ I p“. It is sufficient to show that
n s 3. For this, let Gp be a Sylow p-subgroup and Gq be a Sylow
q-subgroup of G. Since N is a normal p-subgroup, N SLGP.

First, suppose that N i GP’ and consider NGq. If NGq
is not maximal in G, then NGq is supersolvable. Now if Gq is
not maximal in NGq, then Gq is, or is contained in, a fourth
maximal subgroup of G which is subnormal and nilpotent. This implies
that Gq is subnormal (hence normal) in G, a contradiction. Thus
Gq is maximal in NGq and so \N\ I p < p3. On the other hand, if

NGq is maximal in G, then we claim that ‘qu s qz. To show this,

let \Gq‘ 2 q3, and consider the chain:

L2<NL2<NL1<NGq<G ,

where L is maximal in L ,L is maximal in c and L #1.
2 l l q 2

This implies that L2 is contained in a fourth maximal subgroup of

55

G. But fourth maximal subgroups are nilpotent and subnormal, and

so L2 is subnormal in G. Thus L2

q-subgroup of G, which means that there is a non-trivial normal

is contained in every Sylow

q-subgroup of G, a contradiction. Hence \qu s q2. We now have
two possibilities: (a) Suppose \Gq‘ I q2. Let L be a maximal
subgroup of Gd and H be a maximal subgroup of N. If H I 1,
then \N‘ I p and if H I 1, then we form the following maximl

chain:
H <:N <:NL <:NGq < G .

From this we see that H is n-quasinormal in G. Therefore HG

is a subgroup and is clearly maximal in NGq. Consider the chain:
G .< HG <:NG <:G
q q q

If Gq is not maximal in HGq, then Gq is subnormal in G, a con-A
tradiction. Hence Gq must be maximal in .HGq. Since HGq is

supersolvable, we have ‘H\ I p, which shows that \N| I p2 < p3,
the desired conclusion. (b) Now suppose that ‘Gq\ I q and form

the maximal chain:
A < B <:N <:NGq <:G .

If .A I l or B I 1, then \N‘ s p2 and if A I 1, then, as before,

the chain:

<.AG <:NG << G
Gq q q

implies that \A‘ I p, which means that \N\ I p3 and we are finished.

56

Next suppose that N I GP. Then Gq is maximal in G and,
since N is the unique minimal normal subgroup of G, (Gq)G =
(Gq)SG I l and so every third maximal subgroup of Gq is 1. Hence
‘Gq\ s q3. First, suppose that \Gq‘ I q3 and let L be a maximal

subgroup of Gq and K be a maximal subgroup of L. Then

N<NK<NL<G==NGq

is a maximal chain. Let M be a maximal subgroup of N. Since M
is n-quasinormal, MGq is a subgroup. But IMGq *»G and hence
cq = mq by the maximality of sq. Thus M - 1 and 80. m a
\Gp\ I p. Likewise, it can be shown that if ‘Gq‘ I q2, then
\N| s p2 and if ‘Gq‘ I q, then \N‘ 5 p3. This completes the proof
of case 1.

§g§g_z, \G‘ is divisible by three distinct primes p, q,
and r. Let GIK be any prOper factor group of G. If \G/K‘ is
a prime-power, then r(G/K) I 1«< 3; if ‘G/K‘ is divisible by two
distinct primes, then by case 1, r(G/K) s 3; and if ‘G/K‘ is
divisible by all primes p, q, and r, then by induction r(G/K) s 3.
So, as before, we may assume that N is the unique minimal normal
subgroup of G. Without loss of generality, let |N| I p“. We must
show that n s 3. Since G is solvable, there exists a subgroup

M such that M.I GqGr for some Sylow subgroups Gq and Gr' Now

if N I GP, then NM 1‘ G. Hence from the clain:
G <:NG' <:NM[<IG ,
q q

it follows that Gq is maximal in NGq, for otherwise Gq would

be subnormal in G, which is impossible. But NGq is supersolvable.

57

Hence [NGq:Gq] I {N‘ I p. On the other hand, if N I GP, then M

is maximal in G. Since N F\Mb I 1, it follows from the uniqueness
of N that MSG I Mb I 1. Thus every third maximal subgroup of

M is 1. This implies that N is either a third or a second maximal
subgroup of G. Note that N cannot be a first maximal subgroup of
G. First, suppose that N is a third maximal subgroup of G and
let H be a maximal subgroup of N. Then H is n-quasinormal in

G and, since ‘M I GqGr’ HMLI MH is a subgroup. But M is maximal
and HM I G. Hence HM I‘M, which means that H I 1. Thus ‘N| I p
and we are done. Similarly, one can verify that [N‘ s p2 when IN

is second maximal in G. This proves case 2 and the theorem.

BIBLICBRAPHY

10.

11.

12.

13.

14.

R.

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