Let R be an associative ring with identity, \(1 \le \kappa \le \aleph _0\) , and n, k be two positive integers. Denote by \(\mathbb {M}_\kappa (R)\) the ring of all \(\kappa \times \kappa \) square matrices over R if \(\kappa < \aleph _0\) , and the ring of all column-finite infinite matrices over R if \(\kappa = \aleph _0\) . The ring of all upper triangular matrices in \(\mathbb {M}_\kappa (R)\) is denoted by \(\mathbb {T}_\kappa (R)\) . In this paper, we first establish the necessary and sufficient conditions for a matrix to be expressible as a product of matrices whose orders are divisors of k in \(\mathbb {T}_\kappa (R)\) . As an application, we then describe the subgroup of the general linear group of degree n over a field generated by matrices whose orders are divisors of k. Naturally, we also extend this study to the subgroup of the Vershik–Kerov group, that is, the group of infinite matrices with only finitely many nonzero entries below the main diagonal.
