Sets with a self-distributive operation (in the sense of $(a \triangleleft b) \triangleleft c = (a \triangleleft c) \triangleleft (b \triangleleft c)$), in particular quandles, appear in knot and braid theories, Hopf algebra classification, the study of the Yang--Baxter equation, and other areas. An important invariant of quandles is their structure group. The structure group of a finite quandle is known to be either "boring" (free abelian), or "interesting" (non-abelian with torsion). In this paper we explicitly describe all finite quandles with abelian structure group. To achieve this, we show that such quandles are abelian (i.e., satisfy $(a \triangleleft b) \triangleleft c =(a \triangleleft c) \triangleleft b$); present the structure group of any abelian quandle as a central extension of a free abelian group by an explicit finite abelian group; and determine when the latter is trivial. In the second part of the paper, we relate the structure group of any quandle to its 2nd homology group $H_2$. We use this to prove that the $H_2$ of a finite quandle with abelian structure group is torsion-free, but general abelian quandles may exhibit torsion. Torsion in $H_2$ is important for constructing knot invariants and pointed Hopf algebras.

中文翻译：

---

Abelian quandles 和带有阿贝尔结构群的 quandles

具有自分配操作的集合（在 $(a \triangleleft b) \triangleleft c = (a \triangleleft c) \triangleleft (b \triangleleft c)$ 的意义上），特别是 quandles，出现在结和辫子理论中、Hopf 代数分类、Yang--Baxter 方程的研究等领域。quandles 的一个重要不变量是它们的结构群。已知有限四元组的结构群要么是“无聊的”（自由阿贝尔），要么是“有趣的”（有扭转的非阿贝尔）。在本文中，我们明确地描述了所有具有阿贝尔结构群的有限四元组。为了实现这一点，我们证明这样的 quandles 是阿贝尔的（即，满足 $(a \triangleleft b) \triangleleft c =(a \triangleleft c) \triangleleft b$); 将任何阿贝尔四面体的结构群表示为自由阿贝尔群通过显式有限阿贝尔群的中心扩展；并确定后者何时微不足道。在论文的第二部分，我们将任何 quandle 的结构群与其第二个同源群 $H_2$ 联系起来。我们用它来证明具有阿贝尔结构群的有限四元组的$H_2$是无扭转的，但一般的阿贝尔四元组可能会出现扭转。$H_2$ 中的扭转对于构造结不变量和尖头 Hopf 代数很重要。