# Characteristic central subgroup

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: characteristic subgroup and central subgroup
View other subgroup property conjunctions | view all subgroup properties

## Definition

A subgroup $H$ of a group $G$ is termed a characteristic central subgroup if it satisfies both these conditions:

1. $H$ is a characteristic subgroup of $G$.
2. $H$ is a central subgroup of $G$, i.e., it is contained in 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 center central subgroup that is characteristic as a subgroup of the center. follows from center is characteristic and characteristicity is transitive
characteristic subgroup of abelian group Characteristic subgroup of center|FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic subgroup
central subgroup
characteristic central factor
characteristic transitively normal subgroup
abelian characteristic subgroup
abelian normal subgroup