Finite simple permutation groups acting with fixity 4

TL;DR

This paper classifies all finite simple groups that can act faithfully and transitively on a set with exactly four fixed points, linking group actions to geometric concepts like Weierstrass points.

Contribution

It provides a complete classification of finite simple groups with faithful transitive actions of fixity 4, expanding understanding of their symmetry properties.

Findings

Identified all finite simple groups with fixity 4 actions

Detailed the structure of all such group actions

Connected group actions to geometric concepts like Weierstrass points

Abstract

Motivated by the theory of Riemann surfaces and specifically the significance of Weierstrass points, we classify all finite simple groups that have a faithful transitive action with fixity 4, along with details about all possible such actions.