Multivalued Groups and Schur Rings

We study finite involutive multivalued groups associated with Schur rings. Their reduced valency is shown to equal the least common multiple of the sizes of the basic sets of the corresponding ring. We prove that the double-coset group associated with a complemented subgroup of a finite group is isomorphic to a Schur multivalued group over the complement. In particular, every coset multivalued group is a Schur multivalued group. Using affine Schur rings, we construct families of pairwise nonisomorphic commutative multivalued groups over elementary abelian groups. We obtain asymptotic bounds for the sizes of these families and prove that the proportion of double-coset groups in them tends to zero.