Realization of finite groups as isometry groups and problems of minimality

Pedro J. Chocano

Mathematische Nachrichten, 2025, vol. 298, issue 2, 419-426

Abstract: 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.

Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1002/mana.202400287

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bla:mathna:v:298:y:2025:i:2:p:419-426

Ordering information: This journal article can be ordered from
http://www.blackwell ... bs.asp?ref=0025-584X

Access Statistics for this article

Mathematische Nachrichten is currently edited by Robert Denk

More articles in Mathematische Nachrichten from Wiley Blackwell
Bibliographic data for series maintained by Wiley Content Delivery ().