Realization of finite groups as isometry groups and problems of minimality
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2024-11-10
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Wiley
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A finite group šŗ is said to be realized by a finite subset š of a Euclideanspace ā š if the isometry group of š is isomorphic to šŗ. We prove that everyfinite group can be realized by a finite subset š ā ā|šŗ| consisting of |šŗ|(|š| + 1)(ā¤ |šŗ|(log 2 (|šŗ|) + 1)) points, where š is a generating system for šŗ. We defineš¼(šŗ) as the minimum number of points required to realize šŗ in ā š for someš. We establish that |š| provides a sharp upper bound for š¼(šŗ) when consider-ing minimal generating sets. Finally, we explore the relationship between š¼(šŗ)and the isometry dimension of šŗ, that is, defined as the least dimension of theEuclidean space in which šŗ can be realized.
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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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P. J. Chocano, Realization of finite groups as isometry groups and problems of minimality, Math. Nachr. (2024), 1ā8. https://doi.org/10.1002/mana.202400287
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