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 Euclidean space $R^n$ if the isometry group of $V$ is isomorphic to $G$. We prove that every finite group can be realized by a finite subset $V\subset R^{\left|G\right|}$ consisting of $\left|G\right|\left(\right|S|+1\left)\right(\le \left|G\right|({\log }_2\left(\right|G\left|\right)+1\left)\right)$ points, where $S$ is a generating system for $G$. We define $\alpha \left(G\right)$ as the minimum number of points required to realize $G$ in $R^m$ for some $m$. We establish that $\left|V\right|$ provides a sharp upper bound for $\alpha \left(G\right)$ when considering minimal generating sets. Finally, we explore the relationship between $\alpha \left(G\right)$ and the isometry dimension of $G$, that is, defined as the least dimension of the Euclidean space in which $G$ can be realized.