Cover for a group

From Groupprops
Jump to: navigation, search
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