Subgroup of abelian normal subgroup

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: subgroup and Abelian normal subgroup
View other such compositions|View all subgroup properties

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: normal subgroup and Abelian normal subgroup
View other such compositions|View all subgroup properties

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: central subgroup and normal subgroup
View other such compositions|View all subgroup properties

Definition

Symbol-free definition

A subgroup of a group is termed a subgroup of Abelian normal subgroup if it satisfies the following equivalent conditions:

1. It is a subgroup of an Abelian normal subgroup.
2. It is a normal subgroup of an Abelian normal subgroup.
3. It is a central subgroup (i.e., it is contained in the center) of a normal subgroup.

Equivalence of definitions

Further information: equivalence of definitions of subgroup of Abelian normal subgroup

Relation with other properties

Stronger properties

• Central subgroup
• Abelian normal subgroup
• 2-subnormal subgroup of least prime order: For full proof, refer: 2-subnormal of least prime order implies subgroup of Abelian normal subgroup

Weaker properties

• Central factor of normal subgroup
• Abelian 2-subnormal subgroup
• 2-subnormal subgroup
• Right-transitively 2-subnormal subgroup