 
Summary: Realizing Finite Groups in Euclidean
Space
September 6, 1999
Michael O. Albertson 1
Department of Mathematics, Smith College, Northampton MA 01063
albertson@math.smith.edu
Debra L. Boutin
Department of Mathematics, Hamilton College, Clinton NY 13323
dboutin@hamilton.edu
Abstract
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
Rn, 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.
1 Introduction
An object X is said to realize the group G if Aut(X) = G. Here Aut(X) is
the set of bijections from X to itself that preserve whatever is essential about
