2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74151We show that for integers k > 1 and n > 2, the diameter of the Cayley graph of SL_n(Z/kZ) associated to a standard two-element generating set, is at most a constant times n^2 ln k. This answers a question of A. Lubotzky concerning SL_n(F_p) and is unexpected because these Cayley graphs do not form an expander family. Our proof amounts to a quick algorithm for finding short words representing elements of SL_n(Z/kZ).11 pages, no figuresGroup TheoryCombinatoricsprimary 20F05; secondary 05C25, 05C35, 20D06text