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On collineation groups generated by Bol reflections. (English) Zbl 0793.51001
A loop $$(L,o)$$ is a (left) Bol loop if the identity $x o(y o(x o z))=(x o(y o x))o z$ holds for all $$x$$, $$y$$, $$z\in L$$. In a 3-net, $$N$$, one can arbitrarily specify the three pencils as being horizontal $$(H)$$, vertical $$(V)$$ and transversal $$(T)$$. For a point $$P\in N$$ let $$P_ h$$, $$P_ v$$ and $$P_ t$$ denote the horizontal, vertical and transversal lines through $$P$$, respectively. A 3-net which can be coordinatized by a Bol loop is called a Bol 3-net and satisfies the condition: For every vertical line $$g$$, the mapping $$\sigma_ g: N\to N$$ defined by $\sigma_ g(P)=(P_ h\cap g)_ t\cap(P_ t\cap g)_ h$ is a collineation of $$N$$, which preserves the pencil $$V$$ and interchanges the pencils $$H$$ and $$T$$. These collineations of the 3-net are called Bol reflections.
This paper investigates the connection between geometric transformation groups and algebraic invariants for Bol 3-nets. The geometric tools are Bol reflections and the group of collineations that they generate. Modulo a subgroup in the left nucleus of the coordinate loop, the corresponding algebraic invariant becomes the group generated by all left multiplications. This group admits a second interpretation as the subgroup of projectivities generated by perspectivities whose carriers are vertical lines. The authors prove that every such projectivity comes from a collineation generated by Bol reflections. They also examine a wide class of Bol 3-nets where this representation is faithful. Finally, the Bol condition is reformulated as a configuration condition for projective planes, which leads to characterizations of Bol planes and Moufang planes.

##### MSC:
 51A20 Configuration theorems in linear incidence geometry 51A25 Algebraization in linear incidence geometry 51F15 Reflection groups, reflection geometries
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