Cover for a group

This article defines a property that can be evaluated for a subgroup family

View a complete list of properties of subgroup families

Definition

A family ${\mathcal {F}}$ of subgroups of a group $G$ is termed a cover for $G$ if it satisfies the following conditions:

• If $H\in {\mathcal {F}}$, and $x\in G$, then $H^{x}\in {\mathcal {F}}$
• if $M$ is a maximal subgroup of $G$ and $y\in M$ has prime power order, then there exists $H\in {\mathcal {F}}$ such that $y\in H$ and $H\cap M\triangleleft H$

Properties

Inductive cover

Further information: inductive cover for a group

A cover of a group is said to be inductive if given any subnormal subgroup of the group, intersecting each member of the cover with that subnormal subgroup gives a cover of that subgroup.

Prime cover

Further information: Prime cover for a group