Weakly regular and regular sets in Minkowski planes
Minkowski planes correspond to sharply 3-transitive sets G of permutations on a set of at least three elements E. The point set of the Minkowski plane \({\mathcal M}(G)\) constructed from G is \({\mathcal P}=E\times E\) and the blocks of \({\mathcal M}(G)\) are the graphs of maps in G; two points (x,y) and (u,v) are called independent if \(x\neq u\) and \(y\neq v\). The known finite Minkowski planes \({\mathcal M}(p^ m,\sigma)\) have order \(p^ m+1,\) where p is prime and m is a positive integer and \(\sigma\) is an automorphism of the Galois field \(GF(p^ m)\), and they are all constructed in this way from two cosets of \(PSL(2,p^ m)\) (depending on \(\sigma\)) in the projective group \(P\Gamma (2,p^ m)\) in its natural action on the projective line \(Gp(p^ m)\cup \infty.\) In this note the authors are concerned with (weakly) regular subsets of \({\mathcal P}\), that is sets of points whose (pointwise) stabilizer in the automorphism group of \({\mathcal M}(G)\) only consists of the identity. They prove that a set in a Minkowski plane of order n containing at least \((n+3)/2\) independent points and having at least two non-independent points is weakly regular. By examining the Minkowski planes \({\mathcal M}(p^ m,\sigma)\) algebraic conditions are derived for a set to be weakly regular in these planes and it is shown that far smaller sets can be weakly regular. In particular, the minimal weakly regular sets in the finite miquelian Minkowski planes (where \(G=PGL(2,p^ m))\) are sets with four points not contained in a block. Examples of regular sets in finite Minkowski planes of order \(\geq 3\) are provided. Further, checking all possibilities of subsets, it follows that the miquelian Minkowski plane of order 2 contains no regular set and the miquelian Minkowski plane of order 5 contains no regular set of independent points. It is also proved that all other finite Minkowski planes of order \(\neq 7\), \(2^ m\), or \(3^ m\) contain a regular set of 5 independent points.
scientific article; zbMATH DE number 4203152