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 (?, ?, ?)  +
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 (?, ?, ?)  +