Products of Conjugacy Classes in the Special Linear Groups

Abstract

This paper is concerned with products of conjugacy classes in the special linear groups SL _n (F). We obtain a sufficient condition for the product of classes to cover SL _n (F), depending on the shape of the rational normal form of matrices in the classes. One consequence of our result is that if the classes 𝒞 _i in SL _n (q) with 1 ≤ i ≤ khave the property that |𝒞 _i | ≥ |SL _n (q)|¹²then 𝒞 _i  = SL _n (q). Our proof also gives a sharper bound on the extended covering number: any kgenerating conjugacy classes 𝒞 _i in SL _n (F) give 𝒞 _i  = SL _n (F), provided that k ≥ 3n + 4.

Key Words:

• Groups
• Conjugacy classes
• Covering numbers

Acknowledgment

Financial support by EPSRC is gratefully acknowledged.

Notes

^#This paper is a part of the Ph.D. Thesis of the first author, written under the supervision of the second author.