ON PRODUCTS OF CONJUGACY CLASSES IN GENERAL LINEAR GROUPS

Let K be a field and n ≥ 3. Let En(K) ≤ H ≤ GLn(K) be an intermediate group and C a noncentral H-class. Define m(C) as the minimal positive integer m such that ∃i1, . . . , im ∈ {±1} such that the product C i1 · · · 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 = ∞.