Abelian-completed 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]

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Definition

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 $HA=G$.

Relation with other properties

Stronger properties

• Cocentral subgroup

Metaproperties

Upward-closedness

This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group
View other upward-closed subgroup properties

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.