Harmonic involutions and circle reflections in Möbius planes (Q884685)

The author continues his investigation of the Möbius planes \({\mathfrak M}\) from [Adv. Geom. 5, No. 4, 657--661 (2005; Zbl 1085.51007)] in which the set \(H\) of harmonic involutions, that is, involutory automorphisms of \(\mathfrak M\) that fix precisely two points, satisfies the following conditions: (i) For any two different points there exists a harmonic involution with these points as fixed points and the only element with 4 fixed points in the group \(\langle H\rangle\) generated by \(H\) is the identity; (ii) for different points \(P\) and \(Q\) the product of any 3 mappings in \(\{\sigma\in H \mid P^\sigma=Q\}\) has order 2; (iii) the translation group of the derived affine translation plane at each point is a finite-dimensional left vector space over its kernel. In [loc. cit.] the author obtained an algebraic representation of such Möbius planes in terms of a field \(K\) of characteristic \(\neq 2\) and finitely many subfields \(F_i\) of degree 2 in \(K\). In the note under review the author characterizes among these Möbius planes those planes that admit an inversion at each of its circles. He shows that this is the case if and only if the Galois group of \(K\) over the intersection of all the subfields \(F_i\) is abelian.