Abelian marginal subgroup

This article describes a property that arises as the conjunction of a subgroup property: marginal 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 marginal subgroup if $H$ is an abelian group (i.e., it is an abelian subgroup of $G$) and $H$ is also a marginal subgroup of $G$.

Examples

• The center in any group is an abelian marginal subgroup.
• In a finite p-group $G$, the socle coincides with $\Omega_1(Z(G))$, the set of elements of order dividing $p$ in the center (see socle equals Omega-1 of center in nilpotent p-group). This is an abelian marginal subgroup.

Examples in small finite groups

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:

	Group part	Subgroup part	Quotient part
Center of M16	M16	Cyclic group:Z4	Klein four-group
Center of direct product of D8 and Z2	Direct product of D8 and Z2	Klein four-group	Klein four-group

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
marginal 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			|FULL LIST, MORE INFO
abelian normal subgroup	abelian and a normal subgroup: invariant under all inner automorphisms			|FULL LIST, MORE INFO