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arXiv · 1902.02154

On embeddings of quandles into groups

Valeriy Bardakov·Timur Nasybullov
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Abstract

In the present paper, we introduce the new construction of quandles. For a group $G$ and its subset $A$ we construct a quandle $Q(G,A)$ which is called the $(G,A)$-quandle and study properties of this quandle. In particular, we prove that if $Q$ is a quandle such that the natural map $Q\to G_Q$ from $Q$ to its enveloping group $G_Q$ is injective, then $Q$ is the $(G,A)$-quandle for an appropriate group $G$ and its subset $A$. Also we introduce the free product of quandles and study this construction for $(G,A)$-quandles. In addition, we classify all finite quandles with enveloping group $\mathbb{Z}^2$.

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Valeriy Bardakov, Timur Nasybullov. 2019-02-06. On embeddings of quandles into groups. https://arxiv.org/abs/1902.02154

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