# 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 partSubgroup partQuotient part
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

Group partSubgroup partQuotient part
Center of M16M16Cyclic group:Z4Klein four-group
Center of direct product of D8 and Z2Direct product of D8 and Z2Klein four-groupKlein 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