Commutator-in-center subgroup

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 commutator-in-center subgroup if we have:

$[G,H]\le Z(H)$

where $[G,H]$ denotes the commutator of two subgroups and $Z(H)$ denotes the center of $H$.

A group has this property as a subgroup of itself if and only if it has nilpotence class two.

Relation with other properties

Stronger properties

• Central subgroup
• Abelian normal subgroup
• Critical subgroup

Weaker properties

• Normal subgroup