  Let $G$ be a finite group acting on a connected open Riemann surface $X$ by
holomorphic automorphisms and acting on a Euclidean space $\mathbb R^n$ $(n\ge
3)$ by orthogonal transformations. We identify a necessary and sufficient
condition for the existence of a $G$-equivariant conformal minimal immersion
$F:X\to\mathbb R^n$. We show in particular that such a map $F$ always exists if
$G$ acts without fixed points on $X$. Furthermore, every finite group $G$
arises in this way for some open Riemann surface $X$ and $n=2|G|$. We obtain an
analogous result for minimal surfaces having complete ends with finite total
Gaussian curvature, and for discrete infinite groups acting on $X$ properly
discontinuously and acting on $\mathbb R^n$ by rigid transformations.
