Abelian subnormal subgroup

This article describes a property that arises as the conjunction of a subgroup property: subnormal 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 subnormal subgroup if $H$ is an abelian group as a group in its own right (or equivalently, an abelian subgroup of $G$) and $H$ is a subnormal subgroup of $G$.

Examples

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:

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
abelian 2-subnormal subgroup				|FULL LIST, MORE INFO
abelian normal subgroup	abelian and a normal subgroup -- invariant under all inner automorphisms	from normal implies subnormal		|FULL LIST, MORE INFO
abelian characteristic subgroup	abelian and a characteristic subgroup -- invariant under all automorphisms	(via normal)		|FULL LIST, MORE INFO
abelian fully invariant subgroup	abelian and a fully invariant subgroup -- invariant under all endomorphisms			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
nilpotent subnormal subgroup				|FULL LIST, MORE INFO