Associated groups of symmetric quandles

TL;DR

This paper studies the structure and properties of associated groups of symmetric quandles, revealing their relationship with underlying quandles and providing group-theoretic characterizations and embeddability criteria.

Contribution

It introduces a group-theoretic framework for associated groups of symmetric quandles and establishes conditions for their embeddability and abelianization.

Findings

Associated groups of symmetric quandles are characterized group-theoretically.

Symmetric quandle embeddability is equivalent to that of its underlying quandle.

The abelianization of associated groups is explicitly determined.

Abstract

In this paper, we investigate the structure of associated groups of symmetric quandles. Among other results, we explore the relationship between the associated group of a symmetric quandle and that of its underlying quandle. We provide a group-theoretic characterization of associated groups of symmetric quandles. Furthermore, we show that a symmetric quandle is embeddable if and only if its underlying quandle is embeddable, and we determine the abelianization of these associated groups.