Collineations of smooth stable planes

Smooth stable planes have been introduced in [4]. We show that every contin uous collineation between two smooth stable planes is in fact a smooth coll ineation. This implies that the group inverted right perpendicular of all c ontinuous collineations of a smooth stable plane is a Lie transformation gr oup on both the set P of points and the set L of lines. In particular, this shows that the point and line sets of a (topological) stable plane L admit at most one smooth structure such that L becomes a smooth stable plane. Th e investigation of central and axial collineations in the case of (topologi cal) stable planes due to R. Lowen ([25], [26], [27]) is continued for smoo th stable planes. Many results of [26] which are only proved for low dimens ional planes (dim L less than or equal to 4) are transferred to smooth stab le planes of arbitrary finite dimension. As an application of these transfe rs we show that the stabilizers inverted right perpendicular (1)([c,c]) and inverted right perpendicular (1)([A,A]) (see (3.2) Notation) are closed, s imply connected, solvable subgroups of Aut(L) (Corollarp (4.17)). Moreover, we show that inverted right perpendicular([c,c]) is even abelian (Theorem (4.18)). In the closing section we investigate the behaviour of reflections .