Commutative Group Algebras

Let R be a commutative ring with identity. If G is a group, then one can form the group algebra of G over R which we shall denote by R[G]. One can then ask the basic question, how much information about the group G can be deduced from the F-algebra R[G]7 A slightly different formulation is to assume that G and //are groups with R[G] isomorphic to R[H] as /^-algebras and to ask what relations exist between G and H. A discussion of the problem with references can be found in Curtis-Reiner [2, p. 262]. There they prove the early result of Higman [4] that if R is the ring of integers and the groups are finite abelian, then isomorphism of the group algebras implies isomorphism of the groups. Later work of Perlis-Walker [8] and Deskins [3] takes up the problem over fields. One result is that if we take R to be the rational numbers, then the group algebras determine finite abelian groups. There are difficulties in trying to apply the methods of group representations to other situations, particularly if the characteristic of R is finite. However, if the group algebra is commutative, i.e., the group is abelian, then in case R has characteristic p, one has the Frobenius endomorphism of R[G]. In this paper we show that systematic application of this mapping enables one to deduce fairly broad conclusions about abelian groups with isomorphic group algebras over commutative rings. In fact the group modulo its torsion subgroup is completely determined. To deduce conclusions about a /^-primary component of its torsion subgroup however, we must require that R behaves sufficiently well with respect to the prime p. More precisely, one requires that p is not invertible in R. Under this condition, the maximal divisible subgroup and the Ulm invariants of the /^-primary component are determined. A complete statement of results is contained in the theorem in the last section. In the middle two sections we work over fields of various types and then generalize to arbitrary commutative rings later.