Embedding Alexander quandles into groups

TL;DR

This paper proves that every Alexander quandle can be embedded into the conjugation quandle of some group, establishing a universal embedding property for these algebraic structures.

Contribution

It demonstrates that all Alexander quandles, including twisted conjugate ones, can be embedded into groups via conjugation, providing a new perspective on their algebraic relationships.

Findings

Every Alexander quandle embeds into a conjugation quandle of a group.

The result applies to twisted conjugate quandles as well.

Provides a construction for the embedding.

Abstract

For any twisted conjugate quandle $Q$, and in particular any Alexander quandle, there exists a group $G$ such that $Q$ is embedded into the conjugation quandle of $G$.