Polytopes with groups of type PGL_2(q)

TL;DR

This paper identifies the unique regular polytope of rank greater than 3 with automorphism group PGL_2(q), specifically the 4-simplex associated with PGL_2(5), isomorphic to S_5.

Contribution

It establishes the uniqueness of the regular polytope with automorphism group PGL_2(q) for ranks above 3, highlighting the 4-simplex case.

Findings

The 4-simplex is the only such polytope for rank > 3 with automorphism group PGL_2(q).

PGL_2(5) is isomorphic to S_5, linking the polytope to symmetric groups.

Uniqueness of this polytope characterizes the structure of automorphism groups in higher ranks.

Abstract

There exists just one regular polytope of rank larger than 3 whose full automorphism group is a projective general linear group PGL_2(q), for some prime-power q. This polytope is the 4-simplex and the corresponding group is PGL_2(5), which is isomorphic to S_5.