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. Recently, it was proved that every p-Schur ring over an abel-ian group of order p 3 is Schurian. In this paper, we prove that every com-mutative p-Schur ring over a non-abelian group of order p 3 is Schurian. 1. IntroductionLet H be a nite group. We denote by CH the group algebra of H over thecomplex number eld C. For a nonempty subset T ⊆ H, we set T :=P t∈T twhich is treated as an element of CH.A subalgebraA ofthe groupalgebraCH iscalled aSchurringoverH if thereexists a partition Bsets(A) := {T 0 ,T 1 ,...,T r } of H satisfying the followingconditions:(i) {T i | T i ∈ Bsets(A)} is a linear basis of A;(ii) T 0 = {1 H };(iii) T − 1i := {t − | t ∈ T i } ∈ Bsets(A) for all T i ∈ Bsets(A).A Schur ring A over a p-group H is called a p-Schur ring if the size of everyelement in Bsets(A) is a power of p, where p is a prime.Let G be a subgroup of Sym(H) containing the left regular representationof H. We denote by T 0 = {1 H }, T 1 ,...,T r the orbits of the stabilizer G 1 H .The transitivity module V (H,G

COMMUTATIVE p-SCHUR RINGS OVER NON-ABELIAN GROUPS OF ORDER p3

COMMUTATIVE p-SCHUR RINGS OVER NON-ABELIAN GROUPS OF ORDER p³