# Property:Proved in

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## Pages using the property "Proved in"

Showing 25 pages using this property.

2 | |
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2-Sylow subgroup is TI implies it is normal or there is exactly one conjugacy class of involutions + | Gorenstein (302, Chapter 9 (''Groups of even order''), Theorem 1.4, Chapter 9 (''Groups of even order''), Theorem 1.4) + |

3 | |

3-step group implies solvable CN-group + | Gorenstein (401, Lemma 14.1.4, Lemma 14.1.4) + |

A | |

Abelian normal subgroup of core-free maximal subgroup is contranormal implies derived subgroup of whole group is monolith + | Cohn (120, ?, ?) + |

Abelian p-group with indecomposable coprime automorphism group is homocyclic + | Gorenstein (?, ?, ?) + |

Alperin's fusion theorem in terms of well-placed tame intersections + | Gorenstein (284, Theorem 4.5, Chapter 8 (''p-constrained and p-stable groups''), Section 4 (''Groups with subgroups of glauberman type''), Theorem 4.5, Chapter 8 (''p-constrained and p-stable groups''), Section 4 (''Groups with subgroups of glauberman type'')) + |

Analogue of Thompson transitivity theorem fails for abelian subgroups of rank two + | Gorenstein (299, Exercise 8, end of Chapter 8, Exercise 8, end of Chapter 8) + |

Analogue of Thompson transitivity theorem fails for groups in which not every p-local subgroup is p-constrained + | Gorenstein (299, Exercise 7, Chapter 8, Exercise 7, Chapter 8) + |

Any abelian normal subgroup normalizes an abelian subgroup of maximum order + | Gorenstein (274, Theorem 2.6, Section 8.2 (''Glauberman's theorem''), Theorem 2.6, Section 8.2 (''Glauberman's theorem'')) + |

Any class two normal subgroup whose derived subgroup is in the ZJ-subgroup normalizes an abelian subgroup of maximum order + | Gorenstein (278, Theorem 2.9, Chapter 8 (''p-constrained and p-stable groups''), Section 2 (''Glauberman's theorem''), Theorem 2.9, Chapter 8 (''p-constrained and p-stable groups''), Section 2 (''Glauberman's theorem'')) + |

Associative implies generalized associative + | DummitFoote (?, ?, ?) + |

B | |

Brauer's induction theorem + | Serre (75, Theorem 18, Section 10.2, Theorem 18, Section 10.2) + |

Brauer-Fowler inequality relating number of conjugacy classes of strongly real elements and number of involutions + | Gorenstein (306, Chapter 9 (''Groups of even order''), Theorem 1.8, Chapter 9 (''Groups of even order''), Theorem 1.8) + |

Brauer-Fowler theorem on existence of subgroup of order greater than the cube root of the group order + | Gorenstein (303, Chapter 9 (''Groups of even order''), Theorem 1.6, Chapter 9 (''Groups of even order''), Theorem 1.6) + |

Bryant-Kovacs theorem + | HuppertBlackburnII (403, Theorem 13.5, Chapter 13 (''Automorphisms of p-groups''), Theorem 13.5, Chapter 13 (''Automorphisms of p-groups'')) + |

Burnside's theorem on coprime automorphisms and Frattini subgroup + | Gorenstein (?, ?, ?) + |

C | |

Central product decomposition lemma for characteristic rank one + | Gorenstein (?, ?, ?) + |

Centralizer of coprime automorphism in homomorphic image equals image of centralizer + | KhukhroNGA (17, Theorem 1.6.2, Theorem 1.6.2) + |

Centralizer product theorem + | Gorenstein (188, Theorem 3.16, Chapter 5, Section 3 (''p'-automorphisms of p-groups''), Theorem 3.16, Chapter 5, Section 3 (''p'-automorphisms of p-groups'')) + |

Centralizer product theorem for elementary abelian group + | Gorenstein (69, Theorem 3.3, Chapter 3, Section 3 (''Complete reducibility''), Theorem 3.3, Chapter 3, Section 3 (''Complete reducibility'')) + |

Centralizer-commutator product decomposition for finite groups and cyclic automorphism group + | KhukhroNGA (18, Corollary 1.6.4, Corollary 1.6.4) + |

Centralizer-commutator product decomposition for finite nilpotent groups + | Gorenstein (180, Theorem 3.5, Section 5.3 (''p'-automorphisms of p-groups''), Theorem 3.5, Section 5.3 (''p'-automorphisms of p-groups'')) + |

Characteristic implies normal + | RobinsonGT (?, ?, ?) +, KhukhroNGA (?, ?, ?) + |

Characteristic of normal implies normal + | RobinsonGT (28, Section 1.5 (''Characteristic and Fully invariant subgroups''), 1.5.6(iii), Section 1.5 (''Characteristic and Fully invariant subgroups''), 1.5.6(iii)) + |

Characteristic subgroup of Sylow subgroup is weakly closed iff it is normal in every Sylow subgroup containing it + | Gorenstein (255, Theorem 5.1, Chapter 7 (''Fusion, transfer and p-factor groups''), Section 7.5 (''Weak closure and p-normality''), Theorem 5.1, Chapter 7 (''Fusion, transfer and p-factor groups''), Section 7.5 (''Weak closure and p-normality'')) + |

Classification of extraspecial groups + | Gorenstein (?, ?, ?) + |