On collineation groups generated by Bol reflections
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.
scientific article; zbMATH DE number 493204
scientific article