A class of Bol loops with a subgroup of index two

Details A class of Bol loops with a subgroup of index two A class of Bol loops with a subgroup of index two

2007-01-24

arxiv.org/abs/math/0701709v1

proceedings of Loops '03, Prague, published in Comment. Math. Univ. Carolin. 45, no. 2 (2004), 371-381

Mathematics

Group Theory

10 pages

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 $i$, $j\in C_2$. 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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