# Characteristic subgroup of center

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

A subgroup $H$ of a group $G$ is termed a characteristic subgroup of center if $H$ is a central subgroup of $G$ and is a characteristic subgroup of the center of $G$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic subgroup of abelian group the whole group is an abelian group and the subgroup is a characteristic subgroup. any abelian group is its own center. |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic central subgroup the subgroup is a central subgroup and is a characteristic subgroup of the whole group but not necessarily of the center. follows from center is characteristic and characteristicity is transitive. |FULL LIST, MORE INFO
powering-invariant characteristic subgroup characteristic subgroup of center is powering-invariant
quotient-powering-invariant characteristic subgroup characteristic subgroup of center is quotient-powering-invariant
powering-invariant normal subgroup
quotient-powering-invariant subgroup
powering-invariant subgroup
characteristic subgroup
central subgroup
characteristic central factor
characteristic transitively normal subgroup
abelian characteristic subgroup
abelian normal subgroup