# Category:Group property implications

From Groupprops

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 103 pages are in this category, out of 103 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

### 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