On the probability distribution associated to commutator word map in finite groups \rom{2}

TL;DR

This paper investigates the distribution of fiber sizes of the commutator word map in finite groups, establishing conditions under which these sizes are uniform or vary, especially in groups of specific nilpotency classes and conjugacy class structures.

Contribution

It proves that in certain nilpotent groups of class 3 with two conjugacy class sizes, the fiber size set is a singleton, and constructs groups of class 2 with arbitrary fiber size counts.

Findings

In class 3 groups with two conjugacy class sizes, fiber sizes are uniform.

Existence of class 2 groups with any number of fiber sizes.

Characterization of fiber size distributions in specific finite groups.

Abstract

Let $P(G)$ denotes the set of sizes of fibers of non-trivial commutators of the commutator word map. Here, we prove that $|P(G)|=1$, for any finite group $G$ of nilpotency class $3$ with exactlly two conjugacy class sizes. We also show that for given $n\geq 1$, there exists a finite group $G$ of nilpotency class $2$ with exactlly two conjugacy class sizes such that $|P(G)|=n$.