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

Examples/facts

• Jonah-Konvisser abelian-to-normal replacement theorem: This implies that for $k\le 5$ and $p$ odd, the collection of abelian groups of order $p^k$ satisfies a weak normal replacement condition. Note that the theorem actually says that this is a collection of groups satisfying a universal congruence condition, which is a stronger statement.
• Jonah-Konvisser elementary abelian-to-normal replacement theorem: This implies that for $k\le 5$ and $p$ odd, the singleton collection of the elementary abelian group of order $p^k$ satisfies a weak normal replacement condition. Note that the theorem actually says that this is a collection of groups satisfying a universal congruence condition, which is a stronger statement.
• elementary abelian-to-normal replacement theorem for large primes: This implies that for $p>4n-7$, the singleton collection of the elementary abelian group of order $p^n$ satisfies a weak normal replacement condition. The theorem actually states that it is a collection of groups satisfying a strong normal replacement condition.

Tabular form

Collection of groups of order $p^k$	Condition on prime $p$	Condition on $k$	Proof
Elementary abelian group of order $p^k$	Odd	$0\le k\le 5$	Jonah-Konvisser congruence condition on number of elementary abelian subgroups of small prime power order for odd prime
Abelian groups of order $p^k$	Odd	$0\le k\le 5$	Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime
Abelian groups of order $p^k$, exponent dividing $p^d$	Odd	$0\le d\le k\le 5$	Congruence condition on number of abelian subgroups of small prime power order and bounded exponent for odd prime
Groups of order $p^k$, exponent $p$	--	$k<p$	Mann's replacement theorem for subgroups of prime exponent