## Classes of Finite GroupsMany 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advances made, give a new approach to some classic topics, shed light on some fundamental facts that still remain unpublished and present some new subjects of research in the theory of classes of finite, not necessarily solvable, groups.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

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 |

355 | |

367 | |

371 | |

375 | |

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### Common terms and phrases

abelian automorphism charX chief factor chief series class of groups complements composition factor conjugacy class conjugate Consequently Consider contradiction core-free maximal subgroup Corollary cyclic group deﬁned Deﬁnition denote direct product epimorphic exists F-critical F-normaliser F-projector F-subnormal factor of G Fitting class Fitting formation Fitting sets follows formation function G char G containing G-connected G-isomorphic group G group of type Hall system Hence homomorphism implies irreducible isomorphic lattice formation Lemma Let F Let G Let H minimal normal subgroup minimal order monolithic maximal subgroups Moreover nilpotent non-abelian simple group non-trivial normal in G normalises Op(G p-group Proof proper subgroup Proposition prove quotient group result satisﬁes saturated formation Schunck class semidirect product Soc(G soluble groups Statement subgroup of G subgroup-closed saturated subnormal in G subnormal subgroup subsystem of maximal supersoluble supplements Suppose Sylow p-subgroup system of maximal unique minimal normal wreath product X-local formation