Mann's replacement theorem for subgroups of prime exponent

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 fact about::collection of groups satisfying a weak normal replacement condition. Hence, it is also a fact about::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 fact about::collection of groups satisfying a universal congruence condition.

Related facts

 * Abelian-to-normal replacement theorem for prime exponent
 * p-group is either absolutely regular or maximal class or has normal subgroup of exponent p and order p^p
 * Group of exponent p and order greater than p^p is not embeddable in a maximal class group