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Keywords:
Moufang loops; loops $M(G, 2)$; inverse property loops; Bol loops
Moufang loops; loops $M(G, 2)$; inverse property loops; Bol loops
Summary:
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 $i, j\in C_2$. We describe all situations in which the resulting quasigroup is a Bol loop. This paper also corrects an error in P. Vojt\v{e}chovsk'y: On the uniqueness of loops $M(G,2)$.
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