A note on the product of two permutations of prescribed orders

TL;DR

This paper proves a conjecture about the existence of permutation triples with specific orders and product, leading to new results on covers of the complex projective line with bounded degree and ramification.

Contribution

It establishes the existence of permutation triples with prescribed orders and product 1, confirming a conjecture and deriving implications for algebraic covers.

Findings

Proves Kohl's conjecture on permutation triples

Establishes existence of covers with prescribed ramification

Connects permutation properties to algebraic geometry

Abstract

We prove a conjecture by Stefan Kohl on the existence of triples of permutations of bounded degree with prescribed orders and product 1. This result leads to an existence result for covers of the complex projective line with bounded degree and prescribed ramification indices.