On core quandles of groups

We review the definition of a quandle, and in particular core quandle $\mathrm{Core}(G)$ group $G$, which consists underlying set with binary operation $x\lhd y = x y^{-1} x$. This is an involutory i.e., satisfies identity (x\lhd y) y$ addition to other identities defining quandle. Trajectories $(x_i)_{i\in\mathbb{Z}}$ groups quandles (in former context, sequences form $x_i z^i$ where $x,z\in G,$ among characterizations; latter, satisfying $x_{i+1}= x_i\lhd\,x_{i-1})$ are examined. A family necessary conditions for be embeddable noted. Some implications established between holding their quandles. Upper lower bounds obtained on number elements needed generate $G$ finitely generated group. Several questions posed.