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A class of Bol loops with a subgroup of index two. (English) Zbl 1101.20048

With a group \(G\) one can associate eight (not necessarily distinct) binary operations that map \((x,y)\in G\times G\) to \(xy\), \(xy^{-1}\), \(x^{-1}y\), \(x^{-1}y^{-1}\), \(yx\), \(yx^{-1}\), \(y^{-1}x\), \(y^{-1}x^{-1}\). If \(Q\) is a loop and the group \(G\) is its subloop of index two, then one can identify \(Q\) with some \(G\cup\overline G\), where \(x\mapsto\overline x\) is a bijection. In such a setting consider operations \(\circ_i\) on \(G\), \(1\leq i\leq 4\), where \(x\circ_1y=xy\), \(x\circ_2y=\overline{x\overline y}\), \(x\circ_3y=\overline{\overline xy}\) and \(x\circ_4y=\overline x\overline y\). The product in the definition of each of the operations \(\circ_i\) is the product of \(Q\), and one assumes that each of the operations \(\circ_i\) is one of the eight operations induced by \(G\). The paper describes all cases when \(Q\) is a Bol loop. Several new constructions appear.

MSC:

20N05 Loops, quasigroups
20A05 Axiomatics and elementary properties of groups
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