# Collection of groups satisfying a strong normal replacement condition

## Definition

Suppose $\mathcal{S}$ is a finite collection of finite $p$-groups, i.e., groups of prime power order where the prime is $p$. We say that $\mathcal{S}$ satisfies a strong normal replacement condition if it satisfies the following equivalent conditions:

1. For any finite $p$-group $P$ that contains a subgroup $H$ isomorphic to an element of $\mathcal{S}$, $P$ contains a normal subgroup $K$, also isomorphic to an element of $\mathcal{S}$, such that $K$ is contained in the normal closure of $H$ in $P$.
2. For any finite $p$-group $P$ that contains a 2-subnormal subgroup $H$ isomorphic to an element of $\mathcal{S}$, $P$ contains a normal subgroup $K$, also isomorphic to an element of $\mathcal{S}$, such that $K$ is contained in the normal closure of $H$ in $P$.# For any finite $p$-group $Q$ and normal subgroup $P$ of $Q$, if there exists a subgroup of $P$ isomorphic to an element of $\mathcal{S}$, there exists a subgroup of $P$ that is normal in $Q$ and is isomorphic to an element of $\mathcal{S}$.