Abelian verbal subgroup

This article describes a property that arises as the conjunction of a subgroup property: verbal subgroup with a group property (itself viewed as a subgroup property): abelian group
View a complete list of such conjunctions

Definition

A subgroup $H$ of a group $G$ is termed an abelian verbal subgroup if $H$ is an abelian group in its own right (i.e., it is an abelian subgroup of $G$) and $H$ is a verbal subgroup of $G$.

Examples

• In an abelian group, the verbal subgroups are precisely the power subgroups, see verbal subgroup of abelian group.
• In a nilpotent group, second half of lower central series of nilpotent group comprises abelian groups, so all of these are abelian verbal subgroups.
• In a solvable group, the penultimate term of the derived series is an abelian verbal subgroup.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
verbal subgroup of abelian group				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian fully invariant subgroup	abelian and a fully invariant subgroup: invariant under all endomorphisms			|FULL LIST, MORE INFO
abelian characteristic subgroup	abelian and a characteristic subgroup: invariant under all automorphisms			|FULL LIST, MORE INFO
abelian normal subgroup	abelian and a normal subgroup: invariant under all inner automorphisms			|FULL LIST, MORE INFO