Associated groups of symmetric quandles

A symmetric quandle is a quandle equipped with a good involution. In this paper, we study the structure of the associated groups of symmetric quandles. We examine the relationship between the associated group of a symmetric quandle and that of its underlying quandle. As a consequence, we obtain three distinct structural properties of the associated group of the underlying quandle in terms of that of the symmetric quandle. We also provide a group-theoretic characterization of the associated group of a symmetric quandle and compute its abelianization, which coincides with the first symmetric quandle homology. Furthermore, we express the second quandle homology of a quandle in terms of the associated group of the corresponding symmetric quandle. Finally, we show that a symmetric quandle is embeddable in its associated group if and only if its underlying quandle is.