Chiral polyhedra and finite simple groups

TL;DR

This paper demonstrates that most finite non-abelian simple groups can serve as automorphism groups of chiral polyhedra, with specific notable exceptions.

Contribution

It establishes a comprehensive link between finite simple groups and chiral polyhedra, excluding certain classical groups.

Findings

All finite non-abelian simple groups except $PSL_2(q)$, $PSL_3(q)$, $PSU_3(q)$, and $A_7$ act as automorphism groups of chiral polyhedra.

The paper identifies specific groups that do not have this property.

It advances understanding of symmetry groups in polyhedral geometry.

Abstract

We prove that every finite non-abelian simple group acts as the automorphism group of a chiral polyhedron, apart from the groups $PSL_2(q)$, $PSL_3(q)$, $PSU_3(q)$ and $A_7$.