Realization of finite groups as isometry groups and problems of minimality

A finite group G$G$ is said to be realized by a finite subset V$V$ of a Euclidean space Rn$\mathbb {R}^n$ if the isometry group of V$V$ is isomorphic to G$G$. We prove that every finite group can be realized by a finite subset V⊂R|G|$V\subset \mathbb {R}^{|G|}$ consisting of |G|(|S|+1)(≤|G|(log2(|G|)+1))$|G|(|S|+1) (\le |G|(\log _2(|G|)+1))$ points, where S$S$ is a generating system for G$G$. We define α(G)$\alpha (G)$ as the minimum number of points required to realize G$G$ in Rm$\mathbb {R}^m$ for some m$m$. We establish that |V|$|V|$ provides a sharp upper bound for α(G)$\alpha (G)$ when considering minimal generating sets. Finally, we explore the relationship between α(G)$\alpha (G)$ and the isometry dimension of G$G$, that is, defined as the least dimension of the Euclidean space in which G$G$ can be realized.