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

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


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

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

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