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 ~. 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 ~ is contained in all classes which are "well-behaved" in various senses (Most notable of these results is that ~ is contained in all formations which can be locally defined by a system of nonempty formations).
Author(s): Henry G. Bray
Publisher: Polygonal Pub House
Year: 1982
Language: English
Pages: C,viii,231,B
Preface
1 Supersolvable Groups
1. Basic Results
2. Equichained Groups
3. Maximal Subgroups
4. Existence of Subgroups
5. The Generalized Center and Central Series
6. Some Additional Conditions for Supersolvability
7. Supersolvably Embedded Subgroups
8. Weak Normality
2 M-groups
1. Results from Representation Theory
2. Monomial Representanons
3. M-groups
4. Subgroups of M-groups
Exercises
3 CLT and non-CLT Groups
0. Defininons, Comments on Notanon and Terminology, and a Lemma
1. Some Structural Results
2. SBP Groups, BNCLT Groups, and McCarthy Groups; Some Numerical Results
3. Primitive Nonsupersolvable Numbers, Minimal Nonsupersolvable Groups, Pazderski's Theorem, and More Numerical Results
4. Finite non-CLT Groups with Orders of Type 4 or 5
4 Miscellaneous Classes
1. Solvable Linear Groups
2. Groups all of whose Homomorphic Images are CLT-groups
3. Joins of Normal Superso/vable Subgroups
4. Groups whose Lattice of Subgroups is Lower Semi-modular
5. The Classes X and Y
6. Digression: Minimal Permutation Representations
7. Semi-nilpotent Groups
5 Classes of Finite Solvable Groups
1. Formations
2. Fitting Classes
3. Homomorphs and Normal Homomorphs
6 Summary on Closure Properties and Characterizations
1 CLT-groups
2 QCLT-groups
3. Nilpotent-by-Abelian Groups
4. The Sylow Tower Property
5. Supersolvable Groups
6. The Class Y
7. The Class SY
8. LM-groups
Appendixes
Appendix A: Extended Sylow Theory
Appendix B: Chief Factors, Centralizers, and Induced Automorphisms
Appendix C: Various Subgroups
References
Notation
Index
