Abelian-quotient subgroup

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 &lt;math&gt;G&lt;/math&gt; is termed an Abelian-quotient subgroup if it satisfies the following equivalent conditions:


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

Stronger properties

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

Weaker properties

 * Normal subgroup
 * Upward-closedly normal subgroup