Realization of finite groups as isometry groups and problems of minimality

Abstract A finite group is said to be realized by a finite subset of a Euclidean space if the isometry group of is isomorphic to . We prove that every finite group can be realized by a finite subset consisting of points, where is a generating system for . We define as the minimum number of points required to realize in for some . We establish that provides a sharp upper bound for when considering minimal generating sets. Finally, we explore the relationship between and the isometry dimension of , that is, defined as the least dimension of the Euclidean space in which can be realized.