# Berkovich's question on whether a group of prime power order has an outer automorphism of the same prime order

From Groupprops

*This article describes an open problem in the following area of/related to group theory*: p-groups

## Statement

Suppose is a nontrivial group of prime power order that is not cyclic of prime order. Let be the prime. Does have an outer automorphism (i.e., an automorphism that is not an inner automorphism) that has order ?

It is known that the outer automorphism group contains an element of order ; in other words, there exists an outer automorphism whose power is in the inner automorphism group. Berkovich's question asks whether we can in fact find an outer automorphism for which the power is the identity element.