Projective linear groups as automorphism groups of chiral polytopes

Abstract

It is already known that the automorphism group of a chiral polyhedron is never isomorphic to \( PSL \left( 2,q\right) \) or \( PGL \left( 2,q\right) \) for any prime power q. In this paper, we show that \( PSL \left( 2,q\right) \) and \( PGL \left( 2,q\right) \) are never automorphism groups of chiral polytopes of rank at least 5. Moreover, we show that \( PGL \left( 2,q\right) \) is the automorphism group of at least one chiral polytope of rank 4 for every \(q\ge 5\). Finally, we determine for which values of q the group \( PSL \left( 2,q\right) \) is the automorphism group of a chiral polytope of rank 4, except when \(q=p^d\equiv 3\pmod {4}\) where \(d>1\) is not a prime power, in which case the problem remains unsolved.