A finite group H is said to fuse to a finite group G if the class algebra of G is isomorphic to an S-ring over H which is a subalgebra of the class algebra of H. We will also say that G fuses from H. In this case, the classes and characters of H can fuse to give the character table of G. We investigate the case where H is the dihedral group. In many cases, G can be completely determined. In general, G can be proven to have many interesting properties. The theory is developed in terms of S-ring of Schur and Wielandt.

We show that any two split metacyclic groups with the same character tables are isomorphic. We then use this to show that among metacyclic groups that are either 2-groups or are of odd order divisible by at most two primes, that the dihedral and generalized quaternion groups of order 2^n, n = 3, are the only pairs that have the same character tables.

We investigate weak Cayley table isomorphisms, a generalization of group isomorphisms. Suppose G and H are groups. A bijective map phi : G to H is a weak Cayley table isomorphism if it satisfies two conditions:(1) If x is conjugate to y, then phi(x) is conjugate to phi(y); (2) For all x, y in G, phi(xy) is conjugate to phi(x)phi(y).If there exists a weak Cayley table isomorphism between two groups, then we say that the two groups have the same weak Cayley table.This dissertation has two main goals. First, we wish to find sufficient conditions under which two groups have the same weak Cayley table. We specifically study Frobenius groups and groups which satisfy the Camina pair condition. Second, we consider the group of all weak Cayley table isomorphisms between G and itself. We call this group the weak Cayley table group of G and denote it by W(G). Any automorphism of G is an element of W. The inverse map on G is also an element of W. We say that the weak Cayley table group is trivial if it is generated by the set of all automorphisms of G and the inverse map. Stephen Humphries proved that the symmetric groups S_n, the dihedral groups D_{2n} and the free groups F_n (n not equal to 3) all have trivial weak Cayley table groups. We will investigate the weak Cayley table groups of the alternating groups, certain types of Coxeter groups, the projective special linear groups and certain sporadic simple groups.