Abelian-quotient subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

Definition

Symbol-free definition

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

• It contains the commutator subgroup
• It is a normal subgroup and the quotient of the whole group by it is Abelian

Definition with symbols

A subgroup $H$ of a group <math>G</math> is termed an Abelian-quotient subgroup if it satisfies the following equivalent conditions:

• $G' \le H$ where </math>G' = [G,G]</math> is the commutator subgroup of $G$
• $H$ is normal in $G$ and $G/H$ is an Abelian group

Relation with other properties

Stronger properties

• Elementary Abelian-quotient subgroup
• Cyclic quotient-group
• Abelian-completed normal subgroup
• Cocentral subgroup

Weaker properties

• Normal subgroup
• Upward-closedly normal subgroup