Characterization of p-Nilpotent Groups by Monomials of Finite Groups

Let G be a finite group whose order is divisible by a prime number p. Let sigma G be an element of order p of G and let f(X) be the monomial (X sigma)p. Our purpose is to show the following theorem which is a generalization of Thompson's Theorem and Kegel's Theorem. Theorem. If f(Gp')={1}, then G is a p-nilpotent and Op'(G) is nilpotent.