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Realizing Finite Groups in Euclidean Space September 6 , 1999

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 R 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.