Subplanes of a tactical decomposition and singer groups of a projective plane

Abstract

Sufficient and necessary conditions have been obtained for the following: (1) the substructure formed by a member of the partition of points and a member of the partition of lines to be a subplane; (2) the centralizer of a multiplier to be a Baer subplane. We establish the cyclicity of a Sylow 3-subgroup of the multiplier group of an abelian Singer group of square planar order. Sufficient conditions for the existence of a Type II divisor of a Singer group are given. For a Singer group of orderpq, p<q, we prove that if the order of the multiplier group is divisible byp, then the plane will admit a cyclic Singer group.