Finite Soluble Groups

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Walter de Gruyter, 1992 - Mathematics - 891 pages
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Contents

Chapter A Prerequisitesgeneral group theory 1 Groups and subgroupsthe rudiments
1
Groups and homomorphisms
5
Series
7
Direct and semidirect products
9
Gsets and permutation representations
17
Sylow subgroups
21
Commutators
22
Finite nilpotent groups
25
Local formations
356
The theorem of Lubeseder and the theorem of Baer
366
Projectors and local formations
375
Theorems about hypercentral action
386
Chapter V
394
ftnormalizers
400
Connections between normalizers and projectors
409
Chapter VI
426

The Frattini subgroup
30
Soluble groups
34
Theorems of Gaschiitz SchurZassenhaus and Maschke
38
Coprime operator groups
41
Automorphism groups induced on chief factors
44
Subnormal subgroups
47
Primitive finite groups
52
Maximal subgroups of soluble groups
57
The transfer
60
The wreath product
62
Subdirect and central products
73
Extraspecial pgroups and their automorphism groups
77
Automorphisms of abelian groups
83
Chapter B Prerequisitesrepresentation theory 1 Tensor products
90
Projective and injective modules
95
Modules and representations of Kalgebras
101
The structure of a group algebra
111
Changing the field of a representation
120
Induced modules
129
Faithful and simple modules
172
Modules with special properties
182
Group constructions using modules
190
Chapter I
204
The proof of Burnsides ptheorem
210
Hall subgroups
216
System normalizers
235
Pronormal subgroups
241
Normally embedded subgroups
250
Chapter II
262
Some special classes defined by closure properties
271
Chapter III
279
Projectors and covering subgroups
288
Examples
302
Locallydefined Schunck classes and other constructions
321
Projectors in subgroups
328
Connections between Schunck classes and formations
344
Complementation in the lattice
440
Schunck classes with normally embedded projectors
461
Schunck classes with permutable and CAP projectors
471
Chapter VII
477
Supersoluble groups and chief factor rank
485
Primitive saturated formations
497
Strong containment for saturated formations
509
Extreme classes
516
Saturated formations with the coveravoidance property
528
Chapter VIII
535
Normally embedded subgroups are injectors
548
Fischer sets and Fischer subgroups
554
Fitting classesexamples and properties related to injectors
563
Constructions and examples
574
Fischer classes normally embedded and permutable Fitting classes
600
Dominance and some characterizations of injectors
617
Darks constructionthe theme
630
Darks constructionvariations
647
Chapter X
676
Fitting classes and wreath products
697
Normal Fitting classes
704
The Lausch group
720
Examples of Fitting pairs and Bergers theorem
737
The Lockett conjecture
761
Chapter XI
775
Metanilpotent Fitting classes with additional closure properties
783
Further theory of metanilpotent Fitting classes
799
Fitting class boundaries I
806
Fitting class boundaries II
816
Frattini duals and Fitting classes
824
Appendix a A theorem of Gates and Powell
833
Appendix 3 Frattini extensions
846
Bibliography
855
List of Symbols
871
Index of Names
889
Copyright

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