On Schurity of Finite Abelian Groups

A finite group G is called a Schur group, if any Schur ring over G is associated in a natural way with a subgroup of Sym(G) that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this article, it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any noncyclic abelian Schur group of odd order is isomorphic to Z(3)xZ(3)(k) or Z(3)xZ(3)xZ(p) where k1 and p is a prime. In addition, we prove that Z(2)xZ(2)xZ(p) is a Schur group for every prime p.