On Generators and Representations of the Sporadic Simple Groups

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 α ∈F, M is similar to diag(α·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.

Key Words:

• Eigenvalue multiplicities
• Generation by conjugates
• Irreducible representations
• Sporadic simple groups

2000 Mathematics Subject Classification:

• 20F05
• 20C15
• 20C20
• 20C34
• 20C40

ACKNOWLEDGMENT

The second author was supported by FEMAT, Brazil.

Notes

Communicated by S. Sehgal.