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 ${\mathcal {S}}$.

Relation with other properties

Stronger properties

• Collection of groups satisfying a universal congruence condition

Weaker properties

• Collection of groups satisfying a weak normal replacement condition