# Mann's replacement theorem for subgroups of prime exponent

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## Statement

### Statement in terms of strong normal replacement

Suppose $p$ is a prime and $G$ is a finite $p$-group. In other words, $G$ is a group of prime power order. Suppose $G$ contains a subgroup $H$ of order $p^k$, with $k < p$, and exponent $p$. In other words, $H$ is a small group of prime exponent. Then, $G$ contains a normal subgroup of order $p^k$ and prime exponent, which is contained in the normal closure of $H$ in $G$.

In other words, the collection of groups of order $p^k$ and exponent $p$ is a Collection of groups satisfying a weak normal replacement condition (?). Hence, it is also a Collection of groups satisfying a weak normal replacement condition (?).

### Statement in terms of universal congruence condition

Suppose $p$ is a prime and $G$ is a finite $p$-group. In other words, $G$ is a group of prime power order. Suppose $G$ contains a subgroup $H$ of order $p^k$, with $k < p$, and exponent $p$. In other words, $H$ is a small group of prime exponent.

Then, the number of subgroups of $G$ of order $p^k$ and exponent $p$ is congruent to 1 mod $p$.

In other words, the collection of groups of order $p^k$ and exponent $p$ is a Collection of groups satisfying a universal congruence condition (?).