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Published online by Cambridge University Press: 05 August 2013
Introduction
With respect to the structure of torsion subgroups of integral group rings the following conjecture due to Zassenhaus has been over the last twenty years in the middle of the research. The conjecture may be stated as follows.
(ZC) Let ℤG be the integral group ring of the finite group G. Denote the units of augmentation 1 by V(ℤG) and let H be a subgroup of V(ℤG) of the same order as G. Then there exists a central automorphism σ of ℤG with σ(G) = H.
The conjecture is also of interest for more general coefficient rings than ℤ. We say that (ZC) holds for a group ring RG, if the content of the conjecture holds in RG. It has been shown by Roggenkamp and Scott that (ZC) does not hold for any finite group [9], [13]. But, if (ZC) is true, it gives a strong answer to the isomorphism problem of integral group rings. For a recent survey about the Zassenhaus conjecture and related questions we refer to [6]. For the newest developments with respect to the isomorphism problem see [2], [3] rsp.
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