## Abstract

The study of the symmetries of geometric objects has interested generations of mathematicians for hundreds of years, and groups are intended to analyze symmetries of such objects. In fact, a symmetry is merely a self-equivalence of the object, and groups occur concretely in most instances as a family of self-equivalences (automorphisms) of some object such as a topological space, a manifold, a vector space, etc. The elements of such groups are referred to as transformations. We discuss the rudiments of the theory of topological transformation groups in the first section. The second section concerns with geometric motions of the Euclidean spaces. We give here a geometric meaning to the notion of “proper rigid motions” of \({\mathbb {R}}^n\); in particular, we define rotations of \({\mathbb {R}}^n\) and show that each element of the group *SO*(*n*) determines a rotation (in the new sense).