Petr Vojt\v {e}chovsk\'y
On the uniqueness of loops $M(G,2)$

Comment.Math.Univ.Carolinae 44,4 (2003) 629-635.

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 $i,j\in C_2$. 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 $M(G,2)$.

Keywords: Moufang loops, loops $M(G,2)$, inverse property loops, Bol loops
AMS Subject Classification: 20N05