## 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

### Keywords:

Bol loops; subloops of index two
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