2.2 (1966)

A quasigroup is a groupoid $Q(\cdot)$ in which the equations $ax = b$ and $ya = b$ have a unique solution for any $a, b \in Q$. Two quasigroups $Q(\cdot)$ and $Q(\circ)$ are isotopic if there are bijections $\alpha, \beta, \gamma$ of the set $Q$ onto itself such that $x \circ y = \gamma(\alpha x \cdot \beta y)$ for all $x, y \in Q$. It is well-known that all quasigroups that are isotopic to groups form a variety $\mathfrak{G}$. Let $\mathfrak{V}$ be a variety of quasigroups. Characterize the class of groups isotopic to quasigroups in $\mathfrak{G} \cap \mathfrak{V}$. For which identities characterizing $\mathfrak{V}$ is every group isotopic to a quasigroup in $\mathfrak{G} \cap \mathfrak{V}$? Under what conditions on $\mathfrak{V}$ does any group isotopic to a quasigroup in $\mathfrak{G} \cap \mathfrak{V}$ consist of a single element?