Factorizations in finite groups

TL;DR

This paper establishes conditions for unique factorizations in finite groups based on conjugacy classes and applies these results to analyze the structure of Hurwitz spaces related to coverings of the projective line.

Contribution

It provides a necessary and sufficient condition for the uniqueness of factorizations in finite groups with factors from conjugacy classes, and applies this to Hurwitz space component analysis.

Findings

Necessary condition for factorization uniqueness

Sufficient condition when factor counts are large

Application to Hurwitz space component enumeration

Abstract

A necessary condition for uniqueness of factorizations of elements of a finite group $G$ with factors belonging to a union of some conjugacy classes of $G$ is given. This condition is sufficient if the number of factors belonging to each conjugacy class is big enough. The result is applied to the problem on the number of irreducible components of the Hurwitz space of degree $d$ marked coverings of $\mathbb P^1$ with given Galois group $G$ and fixed collection of local monodromies.