Decompositions of finite projective planes (Q2720349)

The paper under review studies the existence of subplanes in general finite projective planes. As is well known, if a projective plane of order \(q\) contains a proper subplane of order \(n\), then either \(q=n^2\) and the subplane is a Baer subplane or \(n^2+n\leq q\). But these conditions are not sufficient to guarantee the existence of a subplane of order \(n\). In fact, all results on the existence of subplanes in general finite projective planes are based on assumptions about the group of automorphisms of the plane in question. NEWLINENEWLINENEWLINEThe author's approach does not use any automorphisms but is based on decompositions of the set \(P\) of points into substructures, primarily these are partitions of \(P\) into sets of collinear points of a given cardinality \(d |q^2+q+1\). The central result is a construction which starts from a proper subplane of order \(r+1\) with \(0<r<d\) and yields a decomposition into sets of \(d\) collinear points each. The decompositions obtained in this way have an additional combinatorial property, and conversely each decomposition with this property gives a subplane of order \(r\).NEWLINENEWLINENEWLINEHence, the author obtained a combinatorial criterion on the existence of subplanes. In particular, it is interesting to construct partitions into sets of collinear points of a given size, and the author proves some theorems of this type.