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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.

51A20 Configuration theorems in linear incidence geometry
51A25 Algebraization in linear incidence geometry
51F15 Reflection groups, reflection geometries
Full Text: DOI
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