Associated groups of symmetric quandles

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.