Conjugation groups and structure groups of quandles

Abstract Quandles are algebraic structures showing up in different mathematical contexts. A group 𝐺 with the conjugation operation forms a quandle, Conj ⁡ ( G ) \operatorname{Conj}(G) . In the opposite direction, a group As ⁡ ( Q ) \operatorname{As}(Q) can be constructed out of any quandle 𝑄. We explore As ⁡ ( Conj ⁡ ( G ) ) \operatorname{As}(\operatorname{Conj}(G)) for a group 𝐺 admitting a presentation with only conjugation and power relations. Symmetric groups S n S_{n} are typical examples. For such groups, we show that As ⁡ ( Conj ⁡ ( G ) ) \operatorname{As}(\operatorname{Conj}(G)) injects into G × Z m G\times\mathbb{Z}^{m} , where 𝑚 is the number of conjugacy classes of 𝐺. From this, we deduce information about the torsion, centre, and derived group of As ⁡ ( Conj ⁡ ( G ) ) \operatorname{As}(\operatorname{Conj}(G)) . As an application, we compute the second integral quandle homology group of Conj ⁡ ( S n ) \operatorname{Conj}(S_{n}) , and unveil rich torsion therein.