Variety-containing subgroup of group of prime power order

This article describes a property that arises as the conjunction of a subgroup property: variety-containing subgroup with a group property imposed on the ambient group: group of prime power order
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

Definition

A variety-containing subgroup of group of prime power order is a subgroup $H$ of a group of prime power order $P$ satisfying the following equivalent conditions:

1. $H={\mathrm{\Omega }}_k(P)$ for some $k$ (i.e., $H$ is one of the omega subgroups of group of prime power order) and the exponent of $H$ divides $p^k$.
2. $H$ is a variety-containing subgroup of $P$.
3. $H$ is a subhomomorph-containing subgroup of $P$.
4. $H$ is a subisomorph-containing subgroup of $P$.

Relation with other properties

Weaker properties

• Isomorph-free subgroup of group of prime power order
• Characteristic subgroup of group of prime power order
• Isomorph-normal characteristic subgroup of group of prime power order
• Fusion system-relatively strongly closed subgroup
• Fusion system-relatively weakly closed subgroup
• Sylow-relatively strongly closed subgroup
• Sylow-relatively weakly closed subgroup