146

U

# Property:Proved in

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## Pages using the property "Proved in"

Showing 20 pages using this property.

C | |
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Corollary of Thompson transitivity theorem + | Gorenstein (293, Theorem 5.6, Chapter 8, Theorem 5.6, Chapter 8) + |

Corollary of centralizer product theorem for rank at least three + | Gorenstein (292, Lemma 5.5, Chapter 8 (''p-constrained and p-stable groups''), Section 5 (''The Thompson Transitivity Theorem''), Lemma 5.5, Chapter 8 (''p-constrained and p-stable groups''), Section 5 (''The Thompson Transitivity Theorem'')) + |

Criterion for projective representation to lift to linear representation + | IsaacsCT (182, Theorem 11.13, Theorem 11.13) + |

Cyclic Frattini quotient implies cyclic + | Gorenstein (?, ?, ?) + |

D | |

Derived subgroup satisfies ascending chain condition on subnormal subgroups implies subnormal join property + | RobinsonGT (?, ?, ?) + |

Dickson's theorem + | Gorenstein (44, Theorem 8.4, Chapter 2 (''Some basic topics''), Section 2.8 (''Two-dimensional linear and projective groups''), Theorem 8.4, Chapter 2 (''Some basic topics''), Section 2.8 (''Two-dimensional linear and projective groups'')) + |

Dihedral trick + | Gorenstein (301, Theorem 1.1, Chapter 9 (''Groups of even order''), Section 1 (''Elementary properties of involutions''), Theorem 1.1, Chapter 9 (''Groups of even order''), Section 1 (''Elementary properties of involutions'')) + |

E | |

Every Sylow subgroup is cyclic implies metacyclic + | Hall (146, Theorem 9.4.3, Theorem 9.4.3) +, Gorenstein (258, Theorem 6.2, Chapter 7 (''Fusion, transfer and p-factor groups''), Section 7.6 (''elementary applications''), Theorem 6.2, Chapter 7 (''Fusion, transfer and p-factor groups''), Section 7.6 (''elementary applications'')) + |

Every nontrivial subgroup of the group of integers is cyclic on its smallest element + | Artin (?, ?, ?) + |

Exactly n elements of order dividing n in a finite solvable group implies the elements form a subgroup + | Hall (145, Theorem 9.4.1, Theorem 9.4.1) + |

Extraspecial commutator-in-center subgroup is central factor + | Gorenstein (?, ?, ?) + |

F | |

Finite group having at least two conjugacy classes of involutions has order less than the cube of the maximum of orders of centralizers of involutions + | Gorenstein (301, Chapter 9 (''Groups of even order''), Theorem 1.3, Chapter 9 (''Groups of even order''), Theorem 1.3) + |

Finite non-abelian 2-group has maximal class iff its abelianization has order four + | Gorenstein (194, Section 5.4 (''p-groups of small depth''), Theorem 4.5, Section 5.4 (''p-groups of small depth''), Theorem 4.5) + |

First isomorphism theorem + | Artin (?, ?, ?) +, DummitFoote (?, ?, ?) + |

Fixed-point-free automorphism of order four implies solvable + | Gorenstein (?, ?, ?) + |

Fixed-point-free automorphism of order three implies nilpotent + | Gorenstein (?, ?, ?) + |

Fixed-point-free involution on finite group is inverse map + | Gorenstein (?, ?, ?) + |

Focal subgroup of a Sylow subgroup is generated by the commutators with normalizers of non-identity tame intersections + | Gorenstein (251, Theorem 4.1, Chapter 7 (''Fusion, Transfer and p-factor groups''), Section 7.4 (''Theorems of Burnside, Frobenius, and Grün''), Theorem 4.1, Chapter 7 (''Fusion, Transfer and p-factor groups''), Section 7.4 (''Theorems of Burnside, Frobenius, and Grün'')) + |

Focal subgroup theorem + | Gorenstein (?, ?, ?) + |

Frattini-in-center odd-order p-group implies p-power map is endomorphism + | Gorenstein (?, ?, ?) + |