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 ${\gamma }_k(G),k\ge (c+1)/2$ 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:

	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
fully invariant subgroup of abelian group				|FULL LIST, MORE INFO
characteristic direct factor 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