The group of isometries of a self-similar set

We prove that any subgroup of isometries of a Euclidean space can occur as a subgroup of isometries of a self-similar set. Furthermore the isometry group of a planar self-similar set E satisfying the open set condition must be finite, and the possible groups of isometries are restricted by the cardinality of the system of similitudes that generates E. We can prove that these results hold for self-similar sets in R 3 only in the case that the generating similarities are homotheties. Lastly we prove that if a self-similar set is strongly separated, then its group of isometries must be finite.