On Generalized Commutator

TL;DR

This paper explores generalized commutator equations in finite groups, deriving explicit character formulas for the number of solutions, extending previous work and providing new insights into the structure of these groups.

Contribution

It introduces explicit formulas for solutions of generalized commutator equations in finite groups, extending prior research and connecting solution counts to group characters.

Findings

Number of solutions equals characters of the group.

Derived explicit formulas for specific classes of finite groups.

Extended previous results by Pournaki and Sobhani.

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].