Sporadic simple groups and quotient singularities

TL;DR

This paper classifies which sporadic simple groups produce exceptional or weakly-exceptional quotient singularities through their faithful representations, identifying the Hall-Janko and Suzuki groups as unique in this regard.

Contribution

It proves that only the Hall-Janko and Suzuki groups yield exceptional or weakly-exceptional quotient singularities via their faithful representations.

Findings

Hall-Janko group produces exceptional quotient singularities

Suzuki group produces weakly-exceptional quotient singularities

Other sporadic simple groups do not produce such singularities

Abstract

We show that the only sporadic simple group such that some of its faithful representations or some faithful representations of its stem extensions give rise to exceptional (weakly-exceptional but not exceptional, respectively) quotient singularities is the Hall-Janko group (the Suzuki group, respectively).