2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/128123Let $K$ be a compact metrizable group and $\Ga$ be a group of automorphisms of $K$. We first show that each $\ap \in \Ga$ is distal on $K$ implies $\Ga$ itself is distal on $K$, a local to global correspondence provided $\Ga$ is a generalized $\ov{FC}$-group or $K$ is a connected finite-dimensional group. We show that $\Ga$ contains an ergodic automorphism when $\Ga$ is nilpotent and ergodic on a connected finite-dimensional compact abelian group $K$.Dynamical Systems37B05, 37A15 (primary) 22B05, 22C05 (secondary)text