146
U

# Property:Proved in

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## Pages using the property "Proved in"

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2
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 (?, ?, ?)  +