Collection of groups satisfying a weak normal replacement condition

Statement

Suppose $\mathcal{S}$ is a collection of finite $p$-groups, i.e., groups of prime power order where the underlying prime is $p$. We say that $\mathcal{S}$ satisfies a weak normal replacement condition if whenever a finite $p$-group $P$ contains a subgroup isomorphic to an element of $\mathcal{S}$, $P$ also contains a normal subgroup isomorphic to an element of $\mathcal{S}$.

Relation with other properties

Stronger properties

• Collection of groups satisfying a universal congruence condition
• Collection of groups satisfying a strong normal replacement condition