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arXiv · 2210.16763

Generalized Alexander quandles of finite groups and their characterizations

Akihiro Higashitani·Hirotake Kurihara
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Abstract

The goal of this paper is to characterization generalized Alexander quandles of finite groups in the language of the underlying groups. Firstly, we prove that if finite groups $G$ are simple, then the quandle isomorphic classes of generalized Alexander quandles of $G$ one-to-one correspond to the conjugacy classes of the automorphism groups of $G$. This correspondence can be also claimed for the case of symmetric groups. Secondly, we give a characterization of generalized Alexander quandles of finite groups $G$ under some assumptions in terms of $G$. As corollaries of this characterization, we obtain several characterizations in some particular groups, e.g., abelian groups and dihedral groups. Finally, we perform a characterization of generalized Alexander quandles arising from groups with their order up to $15$.

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Akihiro Higashitani, Hirotake Kurihara. 2022-10-30. Generalized Alexander quandles of finite groups and their characterizations. https://arxiv.org/abs/2210.16763

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