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

From Groupprops

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

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