On Generalized Commutator

Abstract: In this paper, 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 $[\cdots[[[x_1,x_2],x_3],$ $x_4],\cdots, x_n] = g$, for some well-known classes of finite groups in terms of orders of the group, its center and its commutator subgroup. This paper is an extension of the works of Pournaki and Sobhani in [22].