Characterization of p-nilpotent groups by monomials of finite groups

摘要

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.