The symmetry groups of attractors for smooth equivariant dynamical systems have been classified when the underlying group of symmetries Gamma is finite. The problems that arise when Gamma is compact but infinite are of a completely different nature. We investigate the case when the connected component of the identity Gamma⁰ is Abelian and show that under fairly mild assumptions on the dynamics, it is typically the case that the symmetry of an -limit set contains the continuous symmetries Gamma⁰. Here, typicality is interpreted in both a topological and probabilistic sense (genericity and prevalence).