On the uniqueness of loops M(G,2)

Details On the uniqueness of loops M(G,2) On the uniqueness of loops M(G,2)

2007-01-24

arxiv.org/abs/math/0701705v1

Comment. Math. Univ. Carolin., 44 (2003), no. 4, 629-635

Mathematics

Group Theory

5 pages, revised, the published version contains an error, see "A class of Bol loops with a subgroup of index two" by P.V. for

Scientific paper

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 $M(G,2)$.

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