Characterization of projective planes of small prime orders (Q1071307)

All known projective planes of prime order are Desarguesian. This naturally leads to the conjecture that all of them are Desarguesian. A weaker form would be that the collineation groups of non-Desarguesian planes of prime order must be relatively small, perhaps solvable. The most important result of this paper is that a projective plane of order 11, 13, or 17 is Desarguesian provided one of the following holds: (a) It admits a collineation group which does not leave any point, line, triangle or proper subplane invariant and contains a non-trivial central collineation. (b) It admits a non-abelian simple group. One of the chief tools is the incidence matrix of the tactical decomposition obtained from the point and line orbits of a collineation group.