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 $\sigma (H)\subseteq H$.

Examples

Examples based on subgroup-defining functions and series

• For a solvable group, the penultimate member of the derived series (i.e., the last member before reaching the trivial subgroup) is an abelian fully invariant subgroup.
• For a nilpotent group, 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.

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
fully invariant subgroup of abelian group	the whole group is an abelian group and the subgroup is a fully invariant subgroup			|FULL LIST, MORE INFO
characteristic direct factor of abelian group	the whole group is an abelian group and the subgroup is a characteristic subgroup as well as a direct factor			|FULL LIST, MORE INFO
abelian verbal subgroup	the subgroup is abelian as well as a verbal subgroup			|FULL LIST, MORE INFO
verbal subgroup of abelian group	the subgroup is a verbal subgroup and the whole group is an 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