On the classification of projective planes of order 15 with a Frobenius group of order 30 as a collineation group (Q1059853)

We prove that: There is no projective plane P of order 15 on which operates a Frobenius group of order 30 as a collineation group if the lines of P satisfy the following property: If two lines \(\ell_ i\) and \(\ell_ j\), \(i\neq j\), of P have three pairs of orbit numbers aa bb cc in common then there is a line \(\ell_ k\), \(i\neq k\neq j\) which contains aa bb cc. Here we found one orbital structure for collineation of order 15 of P and indexing it with 0,1,2 (mod 3) we get no compatible sextuples.