On Certain Schur Rings of Dimension 4

Abstract

In this paper, we consider Schur rings on a finite group G of ordern(n-1) suchthat G has a partition $G = S_0 \cup S_1 \cup S_2 \cup S_3$ with$S_0 = \{ 1\} ,\;\;\left| {S_1 } \right| = n - 1,\;\;\left| {S_2 } \right| = n - 2,\;\;\left| {S_3 } \right| = (n - 1)(n - 2)$. Then Gis characterized as follows. (a) G has subgroups E andH of order n andn-1 respectively, and $S_1 = E - \{ 1\} ,\;S_2 = H - \{ 1\}$, or(b)G has subgroupsK andH(≤ 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 subsetR of sizen-1 satisfying $\widehat R \widehat {R^-1} = (n-1) \widehat {S_0} + \widehat {S_3}$ in the groupalgebraC[G]. Then G is characterized as a collineation groupof a projective plane of order n such that G has five orbits ofpoints of lengthsn(n-1), n, n-1, 1 and 1. In particular, we characterize projective planesof ordern admitting a quasiregular collineation group of order n(n-1)as the case that E and H are normal subgroups ofG.