论文标题
在小组的核心问题上
On core quandles of groups
论文作者
论文摘要
我们回顾了一个Quandle的定义，尤其是组的Core Quandle $ \ mathrm {core}（g）$ g $的定义，该$ g $由$ g $的基础集组成，其中二进制操作$ x \ lhd y = x y = x y^y^y^{ - 1} x $。这是一个不足的困境，即满足身份$ x \ lhd（x \ lhd y）= y $，除了定义了Quandle的其他身份。 Trajectories $(x_i)_{i\in\mathbb{Z}}$ in groups and in involutory quandles (in the former context, sequences of the form $x_i = x z^i$ where $x,z\in G,$ among other characterizations; in the latter, sequences satisfying $x_{i+1}= x_i \ lhd \，x_ {i-1}）$。注意到一个必要条件的家族，可以指出一个可以嵌入在一个组的核心问题中的诱因。在团体中和核心难题中保持身份之间建立了一些含义。上限和下限是在生成Quandle $ \ mathrm {core}（g）$的$ g $的元素所需的元素数量上获得的。提出了几个问题。
We review the definition of a quandle, and in particular of the core quandle $\mathrm{Core}(G)$ of a group $G$, which consists of the underlying set of $G$, with the binary operation $x\lhd y = x y^{-1} x$. This is an involutory quandle, i.e., satisfies the identity $x\lhd (x\lhd y) = y$ in addition to the other identities defining a quandle. Trajectories $(x_i)_{i\in\mathbb{Z}}$ in groups and in involutory quandles (in the former context, sequences of the form $x_i = x z^i$ where $x,z\in G,$ among other characterizations; in the latter, sequences satisfying $x_{i+1}= x_i\lhd\,x_{i-1})$ are examined. A family of necessary conditions for an involutory quandle to be embeddable in the core quandle of a group is noted. Some implications are established between identities holding in groups and in their core quandles. Upper and lower bounds are obtained on the number of elements needed to generate the quandle $\mathrm{Core}(G)$ for $G$ a finitely generated group. Several questions are posed.
