EXPLORING THE STRUCTURE OF A FINITE-GROUP GIVEN AN EQUATION IN ITS IRREDUCIBLE CHARACTERS

What can be said about the structure of a finite group G if an equatio n in its irreducible characters holds? This article deals with the spe cial case, where it is assumed that the following equation holds: chi( alpha)(q) = (i=1)Sigma(k) chi(i), q is a prime number. chi(alpha) is a n element of Irr G = {chi(i)}(i=1)(k). The main result is that wheneve r G' is proper in G then, q = 2 and G/G' is an elementary abelian 2-gr oup. In addition if G' is abelian then G is a split extension of an el ementary abelian 2-group with an elementary abelian 3-group.