Scott's formula and Hurwitz groups

TL;DR

This paper advances the classification of Hurwitz groups by utilizing Scott's formula, explicitly identifying generators for certain finite simple groups within projective general linear groups over algebraically closed fields.

Contribution

It completes the classification of Hurwitz groups among finite simple groups in PGL_n(F) for n ≤ 7, and provides explicit generators for specific groups like G_2(q) and Janko groups.

Findings

Classified all Hurwitz groups in PGL_n(F) for n ≤ 7.

Explicit (2,3,7)-generators for G_2(q), J_1, J_2.

Abstract

This paper continues previous work, based on systematic use of a formula of L. Scott, to detect Hurwitz groups. It closes the problem of determining the finite simple groups contained in $PGL_n(F)$ for $n\leq 7$ which are Hurwitz, where $F$ is an algebraically closed field. For the groups $G_2(q)$, $q\geq 5$, and the Janko groups $J_1$ and $J_2$ it provides explicit $(2,3,7)$-generators.