Nobuo Iiyori

p -SOLVABILITY AND A GENERALIZATION OF PRIME GRAPHS OF FINITE GROUPS

ABSTRACT In [1], we defined generalized prime graphs for a finite group and showed their applications. In this note we also study the structure of these graphs, and then we will present some applications of our results.

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 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 Thompsons Theorem and Kegels Theorem. Theorem. If f(G p′) = {1}, then G is a p-nilpotent and O p′(G) is nilpotent.