Abelian fully invariant subgroup

This article describes a property that arises as the conjunction of a subgroup property: fully invariant 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 fully invariant subgroup or fully invariant abelian subgroup if $H$ is an abelian group as a group in its own right (or equivalently, is an abelian subgroup of $G$) and is also a fully invariant subgroup (or fully characteristic subgroup) of $G$, i.e., for any endomorphism $\sigma$ of $G$, we have $\sigma (H)\subseteq H$.

Facts

• Second half of lower central series of nilpotent group comprises abelian groups: In particular, this means that for a group $G$ of nilpotency class $c$, all the subgroups $\gamma _{k}(G),k\geq (c+1)/2$ are abelian characteristic subgroups.

Relation with other properties

Stronger properties

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

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian characteristic subgroup	abelian and a characteristic subgroup -- invariant under all automorphisms	follows from fully invariant implies characteristic	follows from characteristic not implies fully invariant in finite abelian group	|FULL LIST, MORE INFO
abelian normal subgroup	abelian and a normal subgroup -- invariant under all inner automorphisms	(via abelian characteristic, follows from characteristic implies normal)	follows from normal not implies characteristic in the collection of all groups satisfying a nontrivial finite direct product-closed group property	|FULL LIST, MORE INFO
abelian subnormal subgroup	abelian and a subnormal subgroup			|FULL LIST, MORE INFO