On certain Schur rings of dimension 4

In this paper, we consider Schur rings on a finite group G of order n(n-1) such that G has a partition G=S(0)boolean ORS(1)boolean OR S(2)boolean OR S -3 with S-0={1}, |S-1|= n-1, |S-2|=n-2, |S-3|=(n-1)(n-2). Then G is charact erized as follows. (a) G has subgroups E and H of order n and n-1 respectiv ely, and S-1=E-{1}, S-2=H- {1}, or (b) G has subgroups K and H(less than or equal to K) of order 2(n-1) and n-1 respectively, and S-1=K-H, S-2=H-{1}. In addition assume that G has a subset R of size n-1 satisfying (R) over ca p(R-1) over cap=(n-1)(S-0) over cap+(S-3) over cap in the group algebra C[G ]. Then G is characterized as a collineation group of a projective plane of order n such that G has five orbits of points of lengths n(n-1), n, n-1, 1 and 1. In particular, we characterize projective planes of order n admitti ng a quasiregular collineation group of order n(n-1) as the case that E and H are normal subgroups of G.