# Odd-order p-group has coprime automorphism-faithful characteristic class two subgroup of prime exponent

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This article gives the statement, and possibly proof, of a particular subgroup of kind of subgroup in a finite group being coprime automorphism-faithful. In other words, any non-identity automorphism of of the whole group, of coprime order to the whole group, that restricts to the subgroup, restricts to a non-identity automorphism of the subgroup.
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## Statement

Suppose $p$ is an odd prime and $P$ is a finite $p$-group (i.e., a group of prime power order). Then, there exists a subgroup $K$ of $P$ satisfying the following conditions:

• $K$ is a Coprime automorphism-faithful characteristic subgroup (?) of $P$: $K$ is characteristic in $P$ and also
"coprime automorphism-faithful subgroup}coprime automorphism-faithful" is not a number.
in $P$.
• $K$ is a group of nilpotence class two.
• $K$ is a group of prime exponent: the exponent of $K$ is $p$.