On the uniqueness of loops M(G,2)

Abstract

Let $G$ be a finite group and $C_2$ the cyclic group of order 2. Consider the 8 multiplicative operations $(x,y)\mapsto (x^iy^j)^k$, 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 $(G\times\{i\})\times(G\times\{j\})$, for $i$, $j\in C_2$. When $G$ is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomorphic, known as loops $M(G,2)$.