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# Property:Proved in

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## Pages using the property "Proved in"

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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 (?, ?, ?) + |

Classification of finite 2-groups of maximal class + | Gorenstein (194, Section 5.4 (''p-groups of small depth''), Theorem 4.5, Section 5.4 (''p-groups of small depth''), Theorem 4.5) + |

Classification of finite p-groups of characteristic rank one + | Gorenstein (?, ?, ?) + |

Classification of finite p-groups of normal rank one + | Gorenstein (?, ?, ?) + |

Classification of finite p-groups of rank one + | Gorenstein (?, ?, ?) + |

Classification of finite p-groups with cyclic maximal subgroup + | Gorenstein (193, Section 5.4 (''pgroups of small depth''), Theorem 4.4, Section 5.4 (''pgroups of small depth''), Theorem 4.4) + |

Classification of finite p-groups with cyclic normal self-centralizing subgroup + | Gorenstein (?, ?, ?) + |

Classification of finite solvable CN-groups + | Gorenstein (402, Theorem 14.1.5, Theorem 14.1.5) + |

Clifford's theorem + | Gorenstein (?, ?, ?) + |

Commutator of finite group with cyclic coprime automorphism group equals second commutator + | KhukhroNGA (18, Corollary 1.6.4(b), Corollary 1.6.4(b)) + |

Commutator of finite nilpotent group with coprime automorphism group equals second commutator + | Gorenstein (181, Theorem 3.6, Section 5.3 (''p'-automorphisms of p-groups''), Theorem 3.6, Section 5.3 (''p'-automorphisms of p-groups'')) + |

Commuting of non-identity elements defines an equivalence relation between prime divisors of the order of a finite CN-group + | Gorenstein (408, Theorem 14.2.5(i), Theorem 14.2.5(i)) + |

Conjugacy class of prime power size implies not simple + | DummitFoote (890, Lemma 7, Section 19.2 (''Theorems of Burnside and Hall''), Lemma 7, Section 19.2 (''Theorems of Burnside and Hall'')) + |

Conjugacy functor whose normalizer generates whole group with p'-core controls fusion + | Gorenstein (282, Theorem 4.1, Theorem 4.1) + |

Core-free and permutable implies subdirect product of finite nilpotent groups + | LennoxStonehewer (217, Theorem 7.1.10(a), Theorem 7.1.10(a)) + |

Core-free permutable subnormal implies solvable of length at most one less than subnormal depth + | LennoxStonehewer (214, Theorem 7.1.5, Theorem 7.1.5) + |

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 (?, ?, ?) + |

G | |

Glauberman type implies ZJ-functor controls fusion + | Gorenstein (282, Theorem 4.1, Chapter 8 (''p-constrained and p-stable groups''), Theorem 4.1, Chapter 8 (''p-constrained and p-stable groups'')) + |

Glauberman's replacement theorem + | Gorenstein (274, Theorem 2.7, Section 8.2 (''Glauberman's theorem''), Theorem 2.7, Section 8.2 (''Glauberman's theorem'')) + |

Glauberman's theorem on intersection with the ZJ-subgroup + | Gorenstein (278, Theorem 2.10, Chapter 8 (''p-constrained and p-stable groups''), Section 8.2 (''Glauberman's theorem''), Theorem 2.10, Chapter 8 (''p-constrained and p-stable groups''), Section 8.2 (''Glauberman's theorem'')) + |

Glauberman-Thompson normal p-complement theorem + | Gorenstein (280, Theorem 3.1, Chapter 8 (''p-constrained and p-stable groups''), Theorem 3.1, Chapter 8 (''p-constrained and p-stable groups'')) + |

Grün's first theorem on the focal subgroup + | Gorenstein (252, Theorem 4.2, Chapter 7 (''Fusion, transfer and p-factor groups''), Section 7.4 (''Theorems of Burnside, Frobenius and Grün, Theorem 4.2, Chapter 7 (''Fusion, transfer and p-factor groups''), Section 7.4 (''Theorems of Burnside, Frobenius and Grün) + |

H | |

Hall retract implies order-conjugate + | Gorenstein (221, Theorem 2.1(ii), Chapter 6 (''Solvable and pi-solvable groups''), Section 6.2 (''The Schur-Zassenhaus theorem''), Theorem 2.1(ii), Chapter 6 (''Solvable and pi-solvable groups''), Section 6.2 (''The Schur-Zassenhaus theorem'')) + |

Hall subgroups exist in finite solvable group + | DummitFoote (?, ?, ?) + |

Higman's theorem on automorphism of prime order of Lie ring + | KhukhroPAut (83, Section 7.2 (''Combinatorial consequences''), Section 7.2 (''Combinatorial consequences'')) + |

I | |

Intersection of subgroups is subgroup + | DummitFoote (62, Section 2.4 (''Subgroups generated by subsets of a group''), Proposition 8, Section 2.4 (''Subgroups generated by subsets of a group''), Proposition 8) +, RobinsonGT (8, Section 1.3 (''Intersections and joins of subgroups''), Proposition 1.3.2, Section 1.3 (''Intersections and joins of subgroups''), Proposition 1.3.2) +, RobinsonGT (48, 3.3.4, 3.3.4) + |

Invariant special subgroup lemma + | Gorenstein (?, ?, ?) + |

Inverse map is involutive + | DummitFoote (?, ?, ?) + |

Involutions are either conjugate or have an involution centralizing both of them + | Gorenstein (301, Chapter 9 (''Groups of even order''), Theorem 1.2, Chapter 9 (''Groups of even order''), Theorem 1.2) + |

J | |

Join of abelian subgroups of maximum order in Sylow subgroup is characteristic in every p-subgroup containing it + | Gorenstein (271, Lemma 2.2(iv), Section 8.2 (''Glauberman's theorem''), Lemma 2.2(iv), Section 8.2 (''Glauberman's theorem'')) + |

Join of subnormal subgroups is subnormal iff their commutator is subnormal + | RobinsonGT (?, ?, ?) + |

Join of two 3-subnormal subgroups may be proper and contranormal + | RobinsonGT (389, ?, ?) + |

K | |

Kreknin's theorem on automorphism of finite order of Lie ring + | KhukhroPAut (83, Section 7.2 (''Combinatorial consequences''), Section 7.2 (''Combinatorial consequences'')) + |

Kreknin's theorem on existence of Kreknin's function + | KhukhroPAut (83, Section 7.1 (''Graded Lie rings''), Section 7.1 (''Graded Lie rings'')) + |

L | |

Lagrange's theorem + | Artin (?, ?, ?) +, DummitFoote (?, ?, ?) + |

Left and right coset spaces are naturally isomorphic + | AlperinBell (?, ?, ?) + |

Left cosets partition a group + | DummitFoote (?, ?, ?) + |

Lemma on containment in p'-core for Thompson transitivity theorem + | Gorenstein (289, Theorem 5.1, Theorem 5.1) + |

M | |

Maschke's averaging lemma for abelian groups + | Gorenstein (?, ?, ?) + |

Maximal Sylow intersection conjugate in the whole group to another subgroup of the Sylow subgroup is conjugate in the normalizer + | Gorenstein (254, Theorem 4.7, Chapter 7 (''Fusion, transfer and p-groups''), Section 4, Theorem 4.7, Chapter 7 (''Fusion, transfer and p-groups''), Section 4) + |

Maximal among abelian normal implies self-centralizing in supersolvable + | Gorenstein (?, ?, ?) + |

Maximal among abelian normal subgroups in p-Sylow subgroup that is not cyclic implies every invariant p'-subgroup is in the p'-core in p-constrained group + | Gorenstein (289, Lemma 5.2, Chapter 8, Lemma 5.2, Chapter 8) + |

Maximal among abelian normal subgroups of Sylow subgroup implies direct factor of centralizer + | Gorenstein (259, Theorem 6.5, Chapter 7 (''Fusion, transfer and p-factor groups'') ,Section 7.6 (''elementary applications''), Theorem 6.5, Chapter 7 (''Fusion, transfer and p-factor groups'') ,Section 7.6 (''elementary applications'')) + |

Maximal permutable implies normal + | LennoxStonehewer (213, Theorem 7.1.1, Theorem 7.1.1) + |

Maximal subgroup has prime power index in finite solvable group + | Gorenstein (?, ?, ?) + |

N | |

Nilpotent derived subgroup implies subnormal join property + | RobinsonGT (?, ?, ?) + |

Normal rank two Sylow subgroup for least prime divisor has normal complement if the prime is odd + | Gorenstein (257, Theorem 6.1, Chapter 7 (''Fusion, transfer and p-factor groups''), section 7.6 (''elementary applications''), Theorem 6.1, Chapter 7 (''Fusion, transfer and p-factor groups''), section 7.6 (''elementary applications'')) + |

Normal subgroup equals kernel of homomorphism + | DummitFoote (?, ?, ?) + |

Normality is not transitive + | AlperinBell (8, ?, ?) +, DummitFoote (91, Section 3.2 (''More on cosets and Lagrange's theorem''), Example (3), Section 3.2 (''More on cosets and Lagrange's theorem''), Example (3)) + |

Normalizer criterion for maximal among abelian subgroups + | Gorenstein (272, Lemma 2.3, Section 8.2 (''Glauberman's theorem''), Lemma 2.3, Section 8.2 (''Glauberman's theorem'')) + |

Number of nth roots of any conjugacy class is a multiple of n + | Hall (136, Theorem 9.1.1, Section 9.1 (''A theorem of Frobenius, Theorem 9.1.1, Section 9.1 (''A theorem of Frobenius) + |

O | |

Odd-order and CN implies solvable + | Gorenstein (409, Theorem 14.3.1, Theorem 14.3.1) + |

Odd-order implies solvable + | Gorenstein (450, Section 16.2 (''Groups of Odd Order''), Section 16.2 (''Groups of Odd Order'')) + |

Omega-1 of abelian p-group is coprime automorphism-faithful + | Gorenstein (?, ?, ?) + |

Omega-1 of maximal among Abelian normal subgroups with maximum rank in odd-order p-group equals omega-1 of centralizer + | Gorenstein (201, Lemma 4.14, Section 5.4, ''p-groups of small depth'', Lemma 4.14, Section 5.4, ''p-groups of small depth'') + |

Omega-1 of odd-order class two p-group has prime exponent + | Gorenstein (?, ?, ?) + |