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 $M(G,2)$.
@article{119417,
author = {Petr Vojt\v echovsk\'y},
title = {On the uniqueness of loops $M(G,2)$},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {44},
year = {2003},
pages = {629-635},
zbl = {1101.20047},
mrnumber = {2062879},
language = {en},
url = {http://dml.mathdoc.fr/item/119417}
}
Vojtěchovský, Petr. On the uniqueness of loops $M(G,2)$. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 629-635. http://gdmltest.u-ga.fr/item/119417/
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