A probabilistic version of a theorem of Kegel

Abstract.

Let G be a finite group and α an automorphism of G of prime order p. It is shown that if the elements x of G satisfy the equation $xx^{\alpha } x^{{\alpha ^{2} }} \cdots x^{{\alpha ^{{p - 1}} }} = 1$ with sufficiently high probability (depending only on p) then G is nilpotent.