# 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 $\mathcal{S}$.

## 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 $k \le (p + 5)/4$ 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 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 $p = 2$ $p = 3$ $p = 5$ $p = 7$ $p = 11$ $p \ge 11$
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 $p^d, 2 \le d \le k$ 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 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