  Let $K$ 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 $K$.
