# Characteristic subgroup of abelian group

This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property imposed on the ambient group: abelian group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

## Definition

A subgroup $H$ of a group $G$ is termed a characteristic subgroup of abelian group if $H$ is a characteristic subgroup of $G$ and $G$ is an abelian group.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
fully invariant subgroup of abelian group the subgroup is a fully invariant subgroup (i.e., invariant under all endomorphisms) and the whole group is an abelian group. fully invariant implies characteristic characteristic not implies fully invariant in finite abelian group |FULL LIST, MORE INFO
verbal subgroup of abelian group (via fully invariant) (via fully invariant) Fully invariant subgroup of abelian group|FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian characteristic subgroup the subgroup is characteristic and abelian as a group in its own right; the whole group need not be abelian. Follows from abelianness is subgroup-closed the trivial subgroup in any nontrivial group. Characteristic central subgroup, Characteristic subgroup of center|FULL LIST, MORE INFO
characteristic central subgroup characteristic subgroup that is contained in the center obvious trivial subgroup in non-abelian group Characteristic subgroup of center|FULL LIST, MORE INFO
subgroup of abelian group the whole group is abelian; no restriction on the subgroup. obvious any non-characteristic subgroup of an abelian group, e.g., a cyclic subgroup in an elementary abelian group on two or more generators. Abelian-extensible automorphism-invariant subgroup, Abelian-potentially characteristic subgroup, Abelian-quotient-pullbackable automorphism-invariant subgroup|FULL LIST, MORE INFO
characteristic subgroup of center the subgroup is contained in the center and is characteristic as a subgroup of the center trivial subgroup in non-abelian group |FULL LIST, MORE INFO
powering-invariant subgroup and also powering-invariant subgroup of abelian group if the group characteristic subgroup of abelian group implies powering-invariant (any non-characteristic subgroup of a finite abelian group) Characteristic subgroup of center, Intermediately powering-invariant subgroup, Powering-invariant characteristic subgroup, Powering-invariant normal subgroup, Powering-invariant subgroup of abelian group, Quotient-powering-invariant characteristic subgroup, Quotient-powering-invariant subgroup|FULL LIST, MORE INFO
characteristic subgroup of nilpotent group the subgroup is characteristic and the whole group is a nilpotent group follows from abelian implies nilpotent follows from nilpotent not implies abelian (we can use the whole group as the subgroup of itself) |FULL LIST, MORE INFO