On the number of solutions of a generalized commutator equation in finite groups

Abstract

We consider some generalized commutator equations in a finite group and show that the number of solutions of such equations are characters of that group. We also obtain explicit formula for this character, considering the equation [� [[[x1, x2] , x3] , x4] , � , xn] = g, for some well-known classes of finite groups in terms of orders of the group, its center and its commutator subgroup. � 2018, Akad�miai Kiad�, Budapest, Hungary.