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.

Examples in small finite groups

Here are some examples of subgroups in basic/important groups satisfying the property:

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

	Group part	Subgroup part	Quotient part
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

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