The Probability Distribution of Word Maps on Finite Groups

TL;DR

This paper explores how probability distributions from word maps reveal structural properties of finite groups, characterizing nilpotent and abelian groups through uniform fiber sizes and distribution patterns.

Contribution

It establishes that probability distributions from word maps can uniquely identify nilpotent and abelian groups, providing new characterizations and answering existing open questions.

Findings

Finite group is nilpotent iff all surjective word maps have uniform fibers.

Probability distributions from word maps can identify nilpotent groups.

Distributions can determine abelian groups up to isomorphism.

Abstract

Word maps provide a wealth of information about finite groups. We examine the connection between the probability distribution induced by a word map and the underlying structure of a finite group. We show that a finite group is nilpotent if and only if every surjective word map has fibers of uniform size. Moreover, we show that probability distributions themselves are sufficient to identify nilpotent groups, and these same distributions can be used to determine abelian groups up to isomorphism. In addition we answer a question of Amit and Vishne.