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

Satisfaction

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
Elementary abelian group of order $p^k$	Odd	$k\le (p+7)/4$	Elementary abelian-to-normal replacement theorem for large primes
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
Abelian groups of order $p^k$, exponent dividing $p^d$	Any	$0\le d\le k\le (p+1)/2$	Glauberman's abelian-to-normal replacement theorem for bounded exponent and half of prime plus one
Groups of order $p^k$, exponent $p$	--	$k<p$	Mann's replacement theorem for subgroups of prime exponent

Dissatisfaction

Collection of groups of order $p^k$	Condition on prime $p$	Condition on $k$	Proof
Klein four-group	$p=2$	$k=2$	Elementary abelian-to-normal replacement fails for Klein four-group
Elementary abelian group of order $p^k$	$p\ge 7$	$k\ge (p+9)/2$	elementary abelian-to-normal replacement fails for half of prime plus nine for prime greater than five
Abelian groups of order $p^k$	$p\ge 7$	$k\ge (p+9)/2$	abelian-to-normal replacement fails for half of prime plus nine for prime greater than five
Groups of order $p^p$, exponent $p$	all $p$	$k=p$

Threshold values

This lists threshold values of $k$: the largest value of $k$ for which the collection of $p$-groups of order $p^k$ satisfying the stated condition satisfies a universal congruence condition. The nature of all these is such that the universal congruence condition is satisfied for all smaller $k$ but for no larger $k$. The between $a$ and $b$ below means that the minimum known value is $a$ and the maximum known value is $b$.

Collection of groups	$p=2$	$p=3$	$p=5$	$p=7$	$p=11$	$p\ge 11$
Abelian groups of order $p^k$	between 3 and 5	between 5 and 13	between 5 and 6	between 5 and 7	between 6 and 9	between $(p+1)/2$ and $(p+7)/2$
Abelian groups of order $p^k$, exponent dividing $p^d,2\le d\le k$	between 2 and 5	between 5 and 13	between 5 and 6	between 5 and 7	between 6 and 9	between $(p+1)/2$ and $(p+7)/2$
Elementary abelian group of order $p^k$	1	between 5 and 13 (?)	between 5 and 6 (?)	between 5 and 7	between 5 and 10	between the greatest integer $\le (p+7)/4$ and $(p+9)/2$
Groups of exponent $p$, order $p^k$	1	at least 2	at least 4	at least 6