Commutator Equations in Finite Groups

TL;DR

This paper derives formulas to count solutions to commutator equations in finite groups using character theory, extending Frobenius' formula, and applies these to analyze probabilities and dihedral groups.

Contribution

It introduces character-based formulas for counting solutions to commutator systems involving triples in finite groups, extending classical results.

Findings

Formulas expressed in terms of irreducible characters

Explicit calculations for dihedral groups

Applications to probability distributions in group equations

Abstract

The problem of finding the number of ordered commuting tuples of elements in a finite group is equivalent to finding the size of the solution set of the system of equations determined by the commutator relations that impose commutativity among any pair of elements from an ordered tuple. We consider this type of systems for the case of ordered triples and express the size of the solution set in terms of the irreducible characters of the group. The obtained formulas are natural extensions of Frobenius' character formula that calculates the number of ways a group element is a commutator of an ordered pair of elements in a finite group. We discuss how our formulas can be used to study the probability distributions afforded by these systems of equations, and we show explicit calculations for dihedral groups.