Fusions of character tables and Schur rings of abelian groups

If the character table of a finite group H satisfies certain conditions, then the classes and characters of H can fuse to give the character table of a group G of the same order. We investigate the case where H is an abelian group. The theory is developed in terms of the S-rings of Schur and Wielandt. We discuss certain classes of p-groups which fuse from abelian groups and give examples of such groups which do not. We also show that a large class of simple groups do not fuse from abelian groups. The methods to show fusion include the use of extensions which are Camina pairs, but other techniques on S-rings are also developed.