Spiegelungsgeometrie in Lie-Gruppen. (Reflection geometry in Lie groups) (Q1074729)

To deal with calculations with reflections, which play an important role in the foundations of geometry, the concept of the so-called Hjelmslev groups had been considered by \textit{F. Bachmann} [Aufbau der Geometrie aus dem Spiegelungsbegriff (2. Aufl. 1973; Zbl 0254.50001); 'Hjelmslev- Gruppen', Vorlesungsausarbeitung (1971) (2. Neudruck 1976; Zbl 0331.50002)]. To make it more group-theoreticalized, the class of Johnsen groups was introduced. A Johnsen Lie group (JLG) is a Lie group generated (as a topological group) by the subset J of its involutions where J is a union of two disjoint non-empty subsets, the ''lines'' and ''points'', satisfying certain conditions. In the article under review, JLG's are classified, and necessary and sufficient conditions for a Lie group to be a JLG are obtained. As corollaries, the motion groups of the Euclidean, the real and complex hyperbolic and the real affine metric planes are characterized (as JLG's).