2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/63769Each topological group $G$ admits a unique universal minimal dynamical system $(M(G),G)$. When $G$ is a non-compact locally compact group the phase space $M(G)$ of this universal system is non-metrizable. There are however topological groups for which $M(G)$ is the trivial one point system (extremely amenable groups), as well as topological groups $G$ for which $M(G)$ is a metrizable space and for which there is an explicit description of the dynamical system $(M(G),G)$. One such group is the topological group $S_\infty$ of all permutations of the integers ${\mathbb Z}$, with the topology of pointwise convergence. We show that $(M(S_\infty),S_\infty)$ is a symbolic dynamical system (hence in particular $M(S_\infty)$ is a Cantor set), and give a full description of all its symbolic factors. Among other facts we show that $(M(G),G)$ (and hence also every minimal $S_\infty$) has the structure of a two-to-one group extension of proximal system and that it is uniquely ergodic.5 pages. The results in this article will be treated fully in an article, written jointly with B. Weiss, to be published in Geometric and Functional Analysis (GAFA)Dynamical Systems22A05, 22A10, 54H20text