Chiral polyhedra arising from almost simple groups with socle \(PSL(2,q)\) (Q314044)

The main theorem of this paper asserts that if \(G\) is an almost simple group with socle isomorphic to \(\text{PSL}_2(q)\) for some \(q\geq 4\), then \(G\) is the automorphism group of some chiral polyhedron if and only if \(G\) is not one of \(\text{PSL}_2(q)\), \(\text{PGL}_2(q)\), \(\text{M}(1,9)\) or \(\text{PSL}_2(4)\cong \text{PGL}_2(5)\). This theorem is a generalization of results in the literature which cover the particular cases when \(G\cong \text{PSL}_2(q)\) and \(G\cong \text{PGL}_2(q)\). An analogous result also exists in the literature for \textit{regular} polyhedra, as opposed to chiral ones. The proof proceeds by rewriting the condition of being an automorphism group of some chiral polyhedron as a statement about the existence of a pair of generating elements with certain special properties. One then combines this with detailed knowledge of the subgroup structure of almost simple groups with socle \(\text{PSL}_2(q)\) to yield the result. A careful case analysis is required.