On generators and representations of the sporadic simple groups

TL;DR

This paper classifies certain projective representations of sporadic simple groups over algebraically closed fields, focusing on matrices with specific cyclic properties, and explores their generation by conjugate elements.

Contribution

It determines irreducible projective representations with almost cyclic matrices and analyzes generation by minimal conjugate sets in sporadic groups.

Findings

Classified irreducible projective representations with almost cyclic matrices.

Identified conditions for generation of sporadic groups by conjugates.

Provided new insights into matrix properties in group representations.

Abstract

In this paper we determine the irreducible projective representations of sporadic simple groups over an arbitrary algebraically closed field F, whose image contains an almost cyclic matrix of prime-power order. A matrix M is called cyclic if its characteristic and minimum polynomials coincide, and we call M almost cyclic if, for a suitable a in F, M is similar to diag(a Id_h, M_1), where M_1 is cyclic and 0 <= h <= n. The paper also contains results on the generation of sporadic simple groups by minimal sets of conjugate elements.