Realization of finite groups as isometry groups and problems of minimality

Resumen

A finite group šŗ is said to be realized by a finite subset š‘‰ of a Euclideanspace ā„ š‘› if the isometry group of š‘‰ is isomorphic to šŗ. We prove that everyfinite group can be realized by a finite subset š‘‰ āŠ‚ ā„|šŗ| consisting of |šŗ|(|š‘†| + 1)(ā‰¤ |šŗ|(log 2 (|šŗ|) + 1)) points, where š‘† is a generating system for šŗ. We defineš›¼(šŗ) as the minimum number of points required to realize šŗ in ā„ š‘š for someš‘š. We establish that |š‘‰| provides a sharp upper bound for š›¼(šŗ) when consider-ing minimal generating sets. Finally, we explore the relationship between š›¼(šŗ)and the isometry dimension of šŗ, that is, defined as the least dimension of theEuclidean space in which šŗ can be realized.

DescripciĆ³n

I wish to express my gratitude to Manuel A. MorĆ³n 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 there

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CitaciĆ³n

P. J. Chocano, Realization of finite groups as isometry groups and problems of minimality, Math. Nachr. (2024), 1ā€“8. https://doi.org/10.1002/mana.202400287
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