Variety-containing subgroup of group of prime power order

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) $$H = \Omega_k(P)$$ for some $$k$$ (i.e., $$H$$ is one of the defining ingredient::omega subgroups of group of prime power order) and the exponent of $$H$$ divides $$p^k$$.
 * 2) $$H$$ is a defining ingredient::variety-containing subgroup of $$P$$.
 * 3) $$H$$ is a defining ingredient::subhomomorph-containing subgroup of $$P$$.
 * 4) $$H$$ is a defining ingredient::subisomorph-containing subgroup of $$P$$.

Weaker properties

 * Stronger than::Isomorph-free subgroup of group of prime power order
 * Stronger than::Characteristic subgroup of group of prime power order
 * Stronger than::Isomorph-normal characteristic subgroup of group of prime power order
 * Stronger than::Fusion system-relatively strongly closed subgroup
 * Stronger than::Fusion system-relatively weakly closed subgroup
 * Stronger than::Sylow-relatively strongly closed subgroup
 * Stronger than::Sylow-relatively weakly closed subgroup