Realizing Finite Groups in Euclidean Space

A set of points W in Euclidean space is said to realize the finite group G if the isometry group of W is isomorphic to G. We show that every finite group G can be realized by a finite subset of some n, with n < G. The minimum dimension of a Euclidean space in which G can be realized is called its isometry dimension. We discuss the isometry dimension of small groups and offer a number of open questions.