# On embeddings of quandles into groups

## TL;DR

This paper introduces a new class of quandles constructed from groups and their subsets, explores their properties, and classifies finite quandles with a specific enveloping group, advancing the algebraic understanding of quandles.

## Contribution

The paper presents a novel construction of quandles from groups and subsets, and characterizes quandles with injective natural maps to their enveloping groups.

## Key findings

- Defined the $(G,A)$-quandle construction.
- Proved characterization of quandles with injective maps to their enveloping groups.
- Classified all finite quandles with enveloping group $\

## 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 $\mathbb{Z}^2$.