Abelian characteristic subgroup

This article describes a property that arises as the conjunction of a subgroup property: characteristic 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 characteristic subgroup if $H$ is abelian as a group (i.e., $H$ is an abelian subgroup of $G$ and also, $H$ is a characteristic subgroup of $G$, i.e., $H$ is invariant under all automorphisms 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:

	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:

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.
• The center of any group is an abelian characteristic subgroup.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
cyclic characteristic subgroup	cyclic group and a characteristic subgroup of the whole group			|FULL LIST, MORE INFO
characteristic central subgroup	central subgroup (i.e., contained in the center) and a characteristic subgroup of the whole group			|FULL LIST, MORE INFO
abelian critical subgroup				|FULL LIST, MORE INFO
characteristic subgroup of center	characteristic subgroup of the center			|FULL LIST, MORE INFO
characteristic subgroup of abelian group	the whole group is an abelian group and the subgroup is characteristic			|FULL LIST, MORE INFO
maximal among abelian characteristic subgroups				|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
abelian normal subgroup	abelian and a normal subgroup -- invariant under all inner automorphisms	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
class two characteristic subgroup	characteristic subgroup that is also a group of nilpotency class two			|FULL LIST, MORE INFO
nilpotent characteristic subgroup	characteristic subgroup that is also a nilpotent group	follows from abelian implies nilpotent	follows from nilpotent not implies abelian	|FULL LIST, MORE INFO
solvable characteristic subgroup	characteristic subgroup that is also a solvable group	(via nilpotent)	(via nilpotent)	|FULL LIST, MORE INFO

Related group properties

• Group in which every abelian characteristic subgroup is central