Recent Progress on the Isomorphism Problem

A long-standing problem, posed first by Graham Higman in 1940 [Hi] and later by Brauer [Br], is the “isomorphism problem for integral group rings.” Given finite groups G and H, is it true that ZG N ZH implies G N H? Many authors have worked on this question but progress has been difficult [Sal. Perhaps the best positive result was that of Whitcomb in 1968 [Wh], who showed that the implication G N H holds for G metabelian. Dade [D] showed there were counterexamples, even in the metabelian case, if Z was replaced by the family of all fields. We began our investigation also looking for counterexamples. Zassenhaus had conjectured, perhaps as early as 1960, a much stronger result, describing in part just how the isomorphism might come about: Observe first that the group of units of SG, for S any commutative ring, has the form Sx XV where V = V(SG) is the group of normalized units (those with augmentation 1) in SG. So not much is lost by restricting attention to normalized unit groups (subgroups of V), and indeed every group basis for SG is isomorphic by projection to a normalized subgroup. This is one of the first observations of Higman’s thesis [Hi], where it was also shown that the elements of normalized finite unit groups in ZG are linearly independent and generate a pure Z-submodule. Zassenhaus (cf. [Za] or [Se]) conjectured that, if H is any normalized finite unit group in ZG with ZG = ZH (that is, H is a normalized group basis for ZG), then H must be conjugate to G by a unit in QG. Note that the conjectured conjugacy is in QG, and not ZG. It turns out that even the dihedral group of order 8 has two normalized group bases for its group ring over Z, which are not conjugate in ZG, but indeed are conjugate in QG [Wh]. Nevertheless, Klaus Roggenkamp and I still felt that the conjecture was much too strong and about six years ago made a first intensive search for counterexamples. The remainder of this paper, except as otherwise noted, is a report of our joint work.