Research of Schur rings obtained from Combinatorial structure

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Let G be a group and let H_1, ・, H_m be mutually disjoint nontrivial subgroups of G.Set S_0={1}, S_i=H_i*{1} (1<less than or equal>I<less than or equal>m) and S_<m+1>=G**_<1<less than or equal>i<less than or equal>m>H_I We say that the subring (*S_o, *S_1, ・, *S_<m+1>) of the group ring C [G] generated by *S_o, *S_1, *S_<m+1> is a Schur ring if the condition (*S_o, *S_1, *S_<m+1>)=C*S_0+C*S_1+・+C*S_<m+1>. In our article "Difference sets relative to disjoint subgroups" we have shown the following :Theorem. Let D be a difference set of a group G with a parameter A relative to disjoint subgroups H_1, ・, H_m of G ; DD^<(-1)>=*D+<lambda bar>(*G-**H_i). Then one of the following occurs.(i) H=H_1*・*H_m is a subgroup of G and D is an ordinary relative difference set in a group G relative to H.(ii) m=2, <lambda bar>=1, *D=n-1, *G=n(n-1), and {*H_1, *H_2}={n-1, n}.(iii) G is an elementary abelian p-group of order of for some prime p and an integer c, H_<m+1>=S_<m+i>*{1} is a subgroup of G of order p^d and a translate of D is a (p^d, *D, lambda)-difference set in H_<m+1>. Moreover *H_1=・=*H_m=*H_<m+1>=p^d.(iv) *G=n^2 and {H_1, ・, H_m} is a partial spread of G.In our article "On Sylow subgroups of abelian affine difference sets (with Agnes Dizon-Garciano)" we have obtained the following result.Theorem. Let D be an abelian affine difference set of order n in a group G.Let p be a prime divisor of n+1 and let r be the p-rank of G.Assume that p divides w+1 for some w>1 such that pi(w)*pi(n). Set_S=(w-1)_<pi0>, where pi_0=pi((w-1, n^2-1)). Then, r<less than or equal>log_p(*G_S+2).