Theorem: Every finite quasi-group $G$ can be imbedded in a quasi-group $H$ generated by a single element.
Let $\varphi$ is the imbedding. Let $\langle h \rangle.$ Let $g_1 \in G$ and $\varphi(g_1) = h^i$ and $g_2 \in G$ and $\varphi(g_2) = h^j$.
Is it true that $g_1g_2 = \varphi^{-1}(h^{i+1})$?
Second question I have in my mind is
What is the size of $H$?