Chocano, Pedro J.2025-02-062025-02-062024-11-10P. J. Chocano, Realization of finite groups as isometry groups and problems of minimality, Math. Nachr. (2024), 1–8. https://doi.org/10.1002/mana.2024002871522-2616 (online)0025-584X (print)https://hdl.handle.net/10115/75377I wish to express my gratitude to Manuel A. Morón for drawing my attention to [1]. After reading this paper, I started to think about a concrete construction, the one that has being obtained in this paper, to solve the realization problem considered thereA 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.enAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Articlehttps://doi.org/10.1002/mana.202400287info:eu-repo/semantics/openAccess