Vol. 15, No. 2, 1965

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Isomorphic groups and group rings

Donald Steven Passman

Vol. 15 (1965), No. 2, 561–583

Let G be a finite group, S a commutative ring with one and S[G] the group ring of G over S. If H is a group with GH then clearly S[G]S[H] where the latter is an S-isomorphism. We study here the converse question: For which groups G and rings S does S[G]S[H] imply that G is isomorphic to H?

We consider first the case where S = K is a field. It is known that if G is abelian then Q[G]Q[H] implies that GH where Q is the field of rational numbers. We show here that this result does not extend to all groups G. 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 rationals. We show here that the group ring R[G] determines the set of normal subgroups of G along with many of the natural operations defined on this set. For example, under the assumption that G is nilpotent, we show that given normal subgroups M and N, the group ring determines the commutator subgroup (M,N). Finally we consider several special cases. In particular we show that if G is nilpotent of class 2 then R[G]R[H] implies GH.

Mathematical Subject Classification
Primary: 20.80
Received: 28 January 1964
Published: 1 June 1965
Donald Steven Passman