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	$0\leq k\leq 5$	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	$k\leq (p+5)/4$	Elementary abelian-to-normal replacement theorem for large primes
Abelian groups of order $p^{k}$	Odd	$0\leq k\leq 5$	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	$0\leq d\leq k\leq 5$	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	$0\leq d\leq k\leq (p+1)/2$	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 $p^{k}$	$p\geq 7$	$k\geq (p+9)/2$	elementary abelian-to-normal replacement fails for half of prime plus nine for prime greater than five
Abelian groups of order $p^{k}$	$p\geq 7$	$k\geq (p+9)/2$	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 weak normal replacement condition. The nature of all these is such that the weak normal replacement 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	$p=2$	$p=3$	$p=5$	$p=7$	$p=11$	$p\geq 11$
Abelian groups of order $p^{k}$	between 4 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 $p^{d},2\leq d\leq k$	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 6 and 9	between $(p+1)/2$ and $(p+7)/2$
Groups of exponent $p$, order $p^{k}$	1	at least 2	at least 4	at least 6