On products of conjugacy classes in general linear groups

Abstract

Let K be a field and n ≥ 3. Let E n (K) ≤ H ≤ GL n (K) be an intermediate group and C a noncentral H-class. Define m(C) as the minimal positive integer m such that ∃i 1 , . . . , i m ∈ {±1} such that the product C i 1 . . . C im contains all nontrivial elementary transvections. In this article we obtain a sharp upper bound for m(C). Moreover, we determine m(C) for any noncentral H-class C under the assumption that K is algebraically closed or n = 3 or n = ∞.