On products of matrices of a fixed order

We study products of matrices of fixed orders. We show that if g is an upper triangular matrix, finite or infinite, over a field of q elements, then g   can be expressed as a product of at most four triangular matrices whose orders are divisors of q−1q−1. This result can be applied to the general linear and to the Vershik–Kerov group. We also present some facts about conjugacy of elements of orders dividing q−1q−1.