Characterization of p-Nilpotent Groups by Monomials of Finite Groups

Abstract

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