Abelian-completed subgroup

Symbol-free definition
A subgroup of a group is termed Abelian-completed if there is an Abelian subgroup such that their product is the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed Abelian-completed if there is an Abelian subgroup $$A$$ such that $$HA = G$$.

Stronger properties

 * Cocentral subgroup

Metaproperties
If $$H$$ and $$A$$ generate $$G$$, then so do $$K$$ and $$A$$ for any $$K$$ containing $$H$$. Hence, the property of being Abelian-completed is upward-closed.