Isomorphic groups and group rings

Let © be a finite group, £ a commutative ring with one and S[@] the group ring of © over S. If ξ> is a group with © = £ then clearly S[(S] = S[£>] where the latter is an S-isomorphism.We study here the converse question: For which groups © and rings S does £[©] ^ S[ξ>] imply that © is isomorphic to £)?We consider first the case where S = K is a field.It is known that if © is abelian then Q[@] = Q[ξ>] implies that © = §> where Q is the field of rational numbers.We show here that this result does not extend to all groups ©.In fact by a simple counting argument we exhibit a large set of nonisomorphic p-groups with isomorphic group algebras over all noncharacteristic p fields.Thus for groups in general the only fields if interest are those whose characteristic divides the order of the group.We now let S = R be the ring of integers in some finite algebraic extension of the rationale.We show here that the group ring R[@>] determines the set of normal subgroups of © along with many of the natural operations defined on this set.For example, under the assumption that © is nilpotent, we show that given normal subgroups 3Dΐ and 9ΐ, the group ring determines the commutator subgroup (3JI, 91).Finally we consider several special cases.In particular we show that if © is nilpotent of class 2 then R[(g\ = β[ §] implies © = €>.1* Remarks on group algebras* Recently examples have been given of pairs of groups {©, §} for which K[®] is i£-isomorphic to K[φ] for all fields K whose characteristic does not divide the order of the groups.We show here by a simple counting argument that this is not particularly surprising.This approach was suggested by Professor R. Brauer.We prove THEOREM A. Suppose Q[®] ^ Q[ξ>] where Q is the field of rational numbers.Then for all fields K whose characteristic does not divide | © | = | ξ> |, the order of the groups, we have K[®[ ~ THEOREM B. There exists a set of p B{n) nonisomorphic groups of order p n where B{n) = 2/27 (n 3 -17 n 2 ) which have isomorphic group algebras over all noncharacteristic p fields.