Classes of Finite Groups

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Springer Science & Business Media, Jul 10, 2006 - Mathematics - 396 pages
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Many group theorists all over the world have been trying in the last twenty-five years to extend and adapt the magnificent methods of the Theory of Finite Soluble Groups to the more ambitious universe of all finite groups. This is a natural progression after the classification of finite simple groups but the achievements in this area are scattered in various papers. Our objectives in this book were to gather, order and examine all this material, including the latest advances made, give a new approach to some classic topics, shed light on some fundamental facts that still remain unpublished and present some new subjects of research in the theory of classes of finite, not necessarily solvable, groups.
 

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Contents

Maximal subgroups and chief factors
1
12 A generalisation of the JordanHölder theorem
40
13 Crowns
62
14 Systems of maximal subgroups
73
Classes of groups and their properties
87
Basic properties and results
91
23 Schunck classes and projectors
98
24 Fitting classes Fitting sets and injectors
109
Subgroups of soluble type
205
elementary properties
206
52 Existence criteria
215
53 Projectors of soluble type
224
𝕱subnormality
235
62 𝕱subnormal closure
239
63 Lattice formations
247
64 𝕱subnormal subgroups and 𝕱critical groups
265

25 Fitting formations
118
𝖃local formations
125
31 𝖃local formations
126
32 A generalisation of GaschützLubesederSchmidBaer theorem
144
33 Products of 𝖃local formations
153
34 ωlocal formations
163
Normalisers and prefrattini subgroups
169
41 ℌnormalisers
171
42 Normalisers of groups with soluble residual
179
43 Subgroups of prefrattini type
190
65 Wielandt operators
285
Fitting classes and injectors
309
72 Injective Fitting classes
315
73 Supersoluble Fitting classes
329
74 Fitting sets Fitting sets pairs and outer Fitting sets pairs
339
References
355
List of symbols
367
Index of authors
371
Index
375
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