Mann's replacement theorem for subgroups of prime exponent

This article defines a replacement theorem
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Statement

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 $p^n$, with $n<p$, and exponent $p$. In other words, $H$ is a small group of prime exponent. Then, $G$ contains a normal subgroup of order $p^n$ and prime exponent.

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