Category:Group property implications
This category lists important implications between properties over the context space: group. That is, it says that every group satisfying the first group property also satisfies the second
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Pages in category "Group property implications"
The following 105 pages are in this category, out of 105 total. The count includes redirect pages that have been included in the category. Redirect pages are shown in italics.
A
- Abelian automorphism group implies class two
- Abelian implies ACIC
- Abelian implies every element is automorphic to its inverse
- Abelian implies every subgroup is normal
- Abelian implies nilpotent
- Abelian implies self-centralizing in holomorph
- ACIC implies nilpotent (finite groups)
- Additive group of a field implies characteristic in holomorph
- Algebra group implies power degree group for field size
- Algebraically closed implies simple
- Artinian implies co-Hopfian
- Ascending chain condition on normal subgroups implies Hopfian
- Ascending chain condition on subnormal subgroups implies subnormal join property
- At most n elements of order dividing n implies every finite subgroup is cyclic
- Automorphism group is transitive on non-identity elements implies characteristically simple
C
- Center is normality-large implies every nontrivial normal subgroup contains a cyclic normal subgroup
- Centerless and maximal in automorphism group implies every automorphism is normal-extensible
- Characteristically simple implies CSCFN-realizable
- Class two implies generated by abelian normal subgroups
- Class-inverting automorphism implies every element is automorphic to its inverse
- Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving
- Conjugacy-separable implies every quotient-pullbackable automorphism is class-preserving
- Conjugacy-separable implies residually finite
- Cyclic implies abelian
- Cyclic implies abelian automorphism group
- Cyclic over central implies abelian
- Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group
E
F
- Finite abelian implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group
- Finite implies subnormal join property
- Finite minimal simple implies 2-generated
- Finite normal implies amalgam-characteristic
- Finite simple implies 2-generated
- Finite supersolvable implies subgroups of all orders dividing the group order
- Finitely generated abelian implies Hopfian
- Finitely generated abelian implies residually finite
- Finitely generated abelian is subgroup-closed
- Finitely generated and free implies Hopfian
- Finitely generated and nilpotent implies Hopfian
- Finitely generated and residually finite implies Hopfian
- Finitely generated implies countable
- Finitely generated implies every subgroup of finite index has finitely many automorphic subgroups
- Finitely generated implies finitely many homomorphisms to any finite group
- Finitely generated inner automorphism group implies every locally inner automorphism is inner
- Finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups
- Finitely presented and conjugacy-separable implies solvable conjugacy problem
- Finitely presented and residually finite implies solvable word problem
- Finitely presented implies all homomorphisms to any finite group can be listed in finite time
- Frattini-embedded normal-realizable implies every automorph-conjugate subgroup is characteristic
- Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini
- Free implies every subgroup is descendant
- Free implies residually finite
- Free implies residually nilpotent
- FZ implies finite derived subgroup
- FZ implies generalized subnormal join property
G
L
M
N
- Nilpotent automorphism group implies nilpotent of class at most one more
- Nilpotent derived subgroup implies subnormal join property
- Nilpotent implies center is normality-large
- Nilpotent implies every maximal subgroup is normal
- Nilpotent implies every nontrivial normal subgroup contains a cyclic normal subgroup
- Nilpotent implies every subgroup is subnormal
- Nilpotent implies normalizer condition
- Nilpotent implies solvable
- Noetherian implies Hopfian
- Noetherian implies subnormal join property
- Normal implies modular
- Normalizer condition implies every maximal subgroup is normal
- Normalizer condition implies locally nilpotent
O
P
S
- Schur multiplier of cyclic group is trivial
- Schur multiplier of free group is trivial
- Schur multiplier of Z-group is trivial
- Solvable automorphism group implies solvable of derived length at most one more
- SQ-universal implies no nontrivial identity
- Strongly p-solvable implies Glauberman type for odd p
- Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup
- Supersolvable implies nilpotent derived subgroup