Construction of Hurwitz Spaces and Application to the Regular Inverse Problem

TL;DR

This paper introduces a simplified construction of Hurwitz spaces, generalizes them, and applies this framework to prove regularity properties of certain finite groups related to the inverse Galois problem.

Contribution

It provides a new, simplified construction of Hurwitz spaces and extends their scope, leading to novel results on the regularity of specific finite groups.

Findings

Proves regularity of PSO^+_{n}(F_{p^m}) under certain conditions

Generalizes the concept of Hurwitz spaces beyond previous definitions

Establishes connections between Hurwitz spaces and group regularity results

Abstract

The author give a simple construction of Hurwitz spaces which is defined by Fried and Volklein, and generalize Hurwitz spaces. As a consequence of this construction, the author prove the regularities of the groups PSO^+_{n}(\mathbb F_{p^m}) if p is an odd prime which congruentes with 7 modulo 12, n is an even positive integer grater than 11 and m=1 or p is an odd prime which congruentes with 7 modulo 12, \varphi (p^m-1)/2+1\eqiv n/2 (\mod 2), p^m\equiv 3(\mod 4) and n>\max\{\varphi(p^m-1),7\}.