Geodesic
On the uniqueness of loops $M(G,2)$
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 4, pp. 629-635 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $i,j\in C_2$. 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 $M(G,2)$.
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 $i,j\in C_2$. 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 $M(G,2)$.
Classification : 20N05
Keywords: Moufang loops; loops $M(G, 2)$; inverse property loops; Bol loops 
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     title = {On the uniqueness of loops $M(G,2)$},
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Vojtěchovský, Petr. On the uniqueness of loops $M(G,2)$. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 4, pp. 629-635. http://geodesic.mathdoc.fr/item/CMUC_2003_44_4_a6/