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Between
Nilpotent and
Solvable
iiii
iiilli
Between Nilpotent
& Solvable
Nilpotent and
Solvable
Henry G. Bray
W. E. Deskins
David Johnson
John F. Humphreys
B. M. Puttaswamaiah
Paul Venzke
Gary L. Walls
edited by Michael Weinstein
Polygonal Publishing House
80 Passaic Avenue
Passaic, NJ 07055 USA
Copyright © 1982 by Polygonal Publishing House
All rights of publication reserved
Library of Congress Cataloging in Publication Data
Main entry under title:
Between nilpotent and solvable.
Bibliography: p.
Includes indexes.
1. Finite groups. 2. Solvable groups. 3. Groups,
Nilpotent. I. Bray, Henry G. II. Weinstein, Michael.
QA171.B48 512'.22 82-539
ISBN 0-936428-06-6 AACR2
Manufactured in the United States of America
by Braun-Brumfield
Contents
Preface .. .vii
Chapter 1 Supersolvable groups by W. E. Deskins & Paul Venzke ... 1
1. Basic results ... 1
2. Equichained groups ... 8
3. Maximal subgroups ... 11
4. Existence of subgroups... 13
5. The generalized center and central series ... 18
6. Some additional conditions for supersolvability ... 22
7. Supersolvably embedded subgroups ... 28
8. Weak normality ... 38
Chapter 2 M-groups by B. M. Puttaswamaiah ... 43
1. Results from representation theory ... 43
2. Monomial representations... 51
3. M-groups ... 57
4. Subgroups of M-groups... 64
Chapter 3 CLT and non-QLT groups by Henry G. Bray ... 69
0. Definitions, comments on notation and terminology and a lemma .. .69
1. Some structural results... 71
2. SBP groups, BNCLT groups, and McCarthy groups; some numerical
results ... 73
3. Primitive nonsupersolvable numbers, minimal nonsupersolvable groups,
Pazderski’s theorem, and more numerical results ... 92
4. Finite non-CLT groups with orders of type 4 or 5 ... 108
Chapter 4 Miscellaneous classes by John F. Humphreys and David Johnson
... 116
1. Solvable linear groups ... 116
2. Groups all of whose homomorphic images are CLT-groups ... 122
3. Joins of normal supersolvable subgroups ... 126
4. Groups whose lattice of subgroups is lower semi-modular... 129
5. The classes 9C and ^ ... 132
6. Digression: minimal permutation representations ... 136
7. Semi-nilpotent groups ... 137
Chapter 5 Gasses of finite solvable groups by Gary L. Walls . 143
1. Formations ... 144
2. Fitting classes .. A ll
3. Homomorphs and normal homomorphs ... 184
Chapter 6 Summary on closure properties and characterizations by Michael
Weinstein... 187
1. CLT-groups ... 187
2. QCLT-groups ... 188
3. Nilpotent-by-Abelian groups ... 189
VI
4. The Sylow tower property ... 192
5. Super solvable groups ... 195
6. The class ^ . .. 201
7. The class S ^ ... 204
8. LM-groups... .204
Appendix A Extended Sylow theory by W. E. Deskins & Paul Venzke .. . 206
Appendix B Chief factorSy centralizerSy and induced automorphisms by Michael
Weinstein ... 209
Appendix C Various subgroups by W. E. Deskins, John F. Humphreys, David
Johnson, Paul Venzke, Gary L. Walls, and Michael Weinstein .. .212
1. The Frattini subgroup ^(G) ... 212
2. The Fitting subgroup Fit(G) .. .213
3. Gp(G), Gp<G), and Gp/,p(G) ... 215
4. The p-Frattini subgroup <i>p(G) ... 217
5. G^(G) and the nilpotent residual... 218
6. The hypercenter Z*(G) .. . 220
References ... 222
Notation ... 227
Index ... 229
Preface: What the Reader should Know before
Reading and while Reading this Book
The importance and utility of the distinction between solvable and
nonsolvable in finite group theory can hardly be overestimated. Solvable
groups, built up as they are from Abelian groups by repeated extensions,
still bear some family resemblances to their Abelian ancestors, even if these
resemblances are often vague. Nonsolvable groups, on the other hand, are
so wildly non-Abelian that the theories necessary to study them have to be
radically different from those needed in the solvable case.
This book is about solvable groups. It focusses on particular classes of
finite solvable groups (supersolvable, M-group, etcetera) giving, in each
case, at least the basic results about that class. We have chosen to consider
only classes which contain 91, the class of all finite nilpotent groups. This
choice is not as capricious as it first appears. Several results proven in
Chapter 5 show that 9l is contained in all classes which are “well-behaved”
in various senses (Most notable of these results is that 91 is contained in all
formations which can be locally defined by a system of nonempty forma
tions).
The class 91 itself is not given an exposition in the book. This seemed
unnecessary as the basic results about nilpotent groups are so well-known,
being covered in many texts on general group theory. It is assumed then
that the reader is familiar with the concepts of ascending (upper) central
series, descending (lower) central series, and with the following facts about
nilpotent groups:
THEOREM For a finite group G, the following are equivalent:
(1) G is nilpotent.
(2) All Sylow subgroups of G are normal.
(3) All maximal subgroups of G are normal.
(4) All subgroups of G are subnormal.
(5) If H<G, then H <
(6) G is the direct product of its Sylow subgroups.
THEOREM If H < Z(G) and G/ H is nilpotent, then G is nilpotent.
THEOREM If G is nilpotent and {1} 7^ A < G, then Z{G) Pi 7^ {1}.
THEOREM Finite p-groups are nilpotent.
The class S of finite solvable groups is not assumed to be so
well-known to the reader. Fundamental concepts of the theory of solvable
groups are presented in the appendices. The reader is advised to have a
look at these appendices before starting Chapter 1 and to either work
through the appendix material at that time or at least to familiarize himself
with what the appendices contain, with the intention of returning to them if
and when necessary.
Briefly, let us mention some of the basic concepts of the theory of finite
solvable groups. First there is the idea of a Hall subgroup (a subgroup
whose index and order are coprime) and Phillip Hall’s theorem about the
existence and conjugacy of Hall subgroups in solvable groups. This theorem
and related results are referred to as extended Sylow theory since they
extend (for solvable groups) Sylow’s theorems. An account of this theory is
given in Appendix A.
Appendix B discusses chief factors, covering and avoidance, and the
induced automorphism groups A\xXq{H/K). The importance of these
concepts to the study of solvable groups cannot be stressed too highly.
Indeed, an inspection of the contents of a good many of the theorems
throughout the book will reveal a heavy dependence on these concepts. Let
us, therefore, take a minute to define chief factor and point out its most
important property. If G is a group and H and K are normal subgroups of
G with K < H, then H/ K is called a chief factor (or principal factor) of G
if there is no normal subgroup T of G such that K < T < H. Stated
differently, a chief factor of G is a minimal normal subgroup of a quotient
group of G. The single most important property of chief factors—and
something that the reader should keep in mind throughout the book—is
that chief factors of solvable groups are elementary Abelian /^-groups for
some prime p.
In addition to the above, there is one further isolated fact about solvable
groups the reader should know and keep in mind at all times: namely, in a
finite solvable group, the index of a maximal subgroup must be a prime-
power. This is proven as part of Lemma 1.11 in Chapter 5.
A final point that the reader is asked to keep in mind throughout the
book is that all groups under consideration here are finite and solvable.
This requirement will be dropped occasionally—but only occasionally—
and it will be painfully obvious where this is being done (e.g., in theorems,
such as Theorem 2.3 of Chapter 1, whose point it is to establish that a
group meeting certain conditions is solvable). Other than these few occa
sions then, all groups considered are finite and solvable. Despite this
blanket assumption, it has seemed desirable to explicitly use the phrase
“finite solvable” whenever it was wished to emphasize that the group under
consideration was such. As this was done for emphasis, it does not appear
in every case.
Michael Weinstein
