Some Results on Finite Groups Whose Order Contains a Prime to the First Power

Abstract

In a previous investigation [1], the author has studied finite groups of an order g = pg0 where p is a prime and g0 an integer not divisible by p. This work has been continued by H. F. Tuan [5]. Let t denote the number of conjugate classes of which consist of element of order p. Tuan dealt with the groups for which t≦2 and which have a faithful representation of degree less than p - 1. We shall assume here that t≧3. We shall also suppose that does not have a normal subgroup of order p. We state here two results. We shall show (Corollary, Theorem 1) that if / is a faithful irreducible character of of degree n which has T>1 conjugates over the field of the g0-th roots of unity, then