Realization of finite groups as isometry groups and problems of minimality

Abstract

A finite group 𝐺 is said to be realized by a finite subset 𝑉 of a Euclideanspace ℝ 𝑛 if the isometry group of 𝑉 is isomorphic to 𝐺. We prove that everyfinite group can be realized by a finite subset 𝑉 ⊂ ℝ|𝐺| consisting of |𝐺|(|𝑆| + 1)(≤ |𝐺|(log 2 (|𝐺|) + 1)) points, where 𝑆 is a generating system for 𝐺. We define𝛼(𝐺) as the minimum number of points required to realize 𝐺 in ℝ 𝑚 for some𝑚. We establish that |𝑉| provides a sharp upper bound for 𝛼(𝐺) when consider-ing minimal generating sets. Finally, we explore the relationship between 𝛼(𝐺)and the isometry dimension of 𝐺, that is, defined as the least dimension of theEuclidean space in which 𝐺 can be realized.