Let $G$ be a finite group and $C_2$ the cyclic group of order 2. Consider the 8 multiplicative operations $(x,y)\mapsto {\left(x^i{y}^j\right)}^k$, where $i,j,k\in \{-1,1\}$. Define a new multiplication on $G\times C_2$ by assigning one of the above 8 multiplications to each quarter $(G\times \{i\left\}\right)\times (G\times \{j\left\}\right)$, for $i,j\in C_2$. If the resulting quasigroup is a Bol loop, it is Moufang. When $G$ is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomorphic, known as loops $M(G,2)$.

If $(G,·)$ is a group, and the operation $(*)$ is defined by $x*y=x·y^{-1}$ then by direct verification $(G,*)$ is a quasigroup which satisfies the identity $(x*y)*(z*y)=x*z$. Conversely, if one starts with a quasigroup satisfying the latter identity the group $(G,·)$ can be constructed, so that in effect $(G,·)$ is determined by its right division operation. Here the analogous situation is examined for a Moufang loop. Subtleties arise which are not present in the group case since there is a choice of defining identities and the identities...