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]

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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
characteristic subgroup
central subgroup
characteristic central factor
characteristic transitively normal subgroup
abelian characteristic subgroup
abelian normal subgroup