On core quandles of groups

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 necessary condition is noted for an involutory quandle to be embeddable in the core quandle of a group. 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.