On the collineation groups of infinite projective and affine planes

A partial plane is a triple Π=(P,L,I) whereP is the set of points,L the set of lines andI?PXL the incidence relation satisfying the axiom that $$p_i {\rm I}\ell _j (i,j = 1,2) implies p_1 = p_2 or \ell _1 = \ell _2 .$$ Using methods of E. MENDELSOHN, Z. HEDRLIN and A. PULTR we prove the followingTHEOREM. Given a subgroup G ofthe collineation group Aut Π ofsome partial plane Π, there is a projective plane Π′such that Πis invariant under the automorphisms of Π′, Aut Π′Π^′=G,and we obtain an isomorphism of Aut Πonto Aut Π′by restriction. Moreover, any 3 points (lines) of Πare collinear (concurrent) in Π iff they are so in Π′. Corollaries of this result improve some of E. Mendelsohn's theorems 6,7].