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We determine the structure of a nonabelian group G of odd order such that some automorphism of G sends exactly (l/p)|G| elements to their cubes, where p is the smallest prime dividing \G\. These groups are close to being abelian in the sense that they either have nilpotency class 2 or have an abelian subgroup of index p.

Abstract.A finite group having all (nontrivial) elements of prime order must be a p-group of exponent p , or a nonnilpotent group of order paq , or it is isomorphic to the simple group A<.

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