A course on integral group rings (Q2707648)

Let \(\mathbb{Z} G\) be the integral group ring of a finite group \(G\). The article is a brief introductory survey on some of the main problems about units of \(\mathbb{Z} G\): the isomorphism problem, the Zassenhaus conjectures and related questions. The isomorphism problem asks whether or not \(\mathbb{Z} G\) determines \(G\) up to isomorphism, whereas the Zassenhaus conjecture, in its strongest form, says that the torsion subgroups of units in \(\mathbb{Z} G\) are trivial up to conjugation in the rational group algebra \(\mathbb{Q} G\). A related problem, the automorphism conjecture, also attributed to H. Zassenhaus, asserts that every normalized automorphism of \(\mathbb{Z} G\) can be written as a composition of an automorphism induced from a group automorphism of \(G\) followed by an inner automorphism of \(\mathbb{Q} G\).NEWLINENEWLINENEWLINEAfter the introductory Section 1 the author gives in Section 2 some basic notions and classical facts about the integral representations of finite groups. Semilocalizations, class groups of \(\mathbb{Z} G\) and the Krull-Schmidt-Azumaya theorem are also considered. Section 3 deals with small group rings, orders, pull-backs and their applications to the study of automorphisms and isomorphisms of integral groups rings. Section 4 begins with basic facts about torsion units in integral group rings followed by consideration of automorphisms of \(\mathbb{Z} G\). The use of the character table automorphisms for the automorphism conjecture is illustrated on the alternatig group \(A_5\), whereas for the Mathieu group \(M_{23}\) modular methods are applied in addition to the character table analysis. The author considers next the normalizer conjecture which is relevant to the isomorphism problem. The conjecture says that the normalizer of \(G\) in the unit group of \(\mathbb{Z} G\) is trivial, i.e. it is generated by \(G\) and the centralizer of \(G\) in the unit group of \(\mathbb{Z} G\). The results by D. B. Coleman, S. Jackowski -- Z. Marciniak and J. Krempa are given and, in connection with the isomorphism problem, results by M. Hertweck, M. Mazur and K. W. Roggenkamp -- A. Zimmermann are mentioned. The relation between the knowledge of the automorphisms of the integral group rings of finite Coxeter groups and the automorphisms of Hecke algebras is also treated. Then the author goes to Hertweck's counterexample to the isomorphism problem. Interesting computations show the use of pullbacks in the construction of units in \(\mathbb{Z} G\). Finally, in Section 5 a survey is given on the considered problems which includes results about particular groups, counterexamples and structural results. The article contains nice illustrative examples which help to understand the basic notions as well as the advanced ideas.NEWLINENEWLINENEWLINEIt should be noted that a statement in Section 5 needs a correction. It is written in the first subsection that the Zassenhaus conjecture for torsion units (ZC 1) is valid for metacyclic groups, referring to an article by \textit{Z. Marciniak, J. Ritter, S. K. Sehgal} and \textit{A. Weiss} [J. Number Theory 25, 340-352 (1987; Zbl 0611.16007)]. According to the reviewer's knowledge, (ZC 1) is not proved for all finite metacyclic groups. The most general result is obtained by \textit{C. Polcino Milies, J. Ritter} and \textit{S. K. Sehgal} [in Proc. Am. Math. Soc. 97, 201-206 (1986; Zbl 0594.16001)] for split metacyclic groups with cyclic factors of relatively prime orders. The article mentioned by the author treats some metabelian groups, which in the metacyclic case gives less generality. The Zassenhaus conjecture (ZC 1) for non-split metacyclic groups remains to be a major open problem in this direction.NEWLINENEWLINENEWLINESince the article has introductory nature, it is hard to expect the consideration of all related aspects. Thus, many interesting results are not mentioned like a series of papers on the automorphism conjecture for wreath products of groups, important results on torsion matrices over integral group rings and on torsion units in integral group rings of infinite groups. One may complement this introduction consulting a short survey by the reviewer [Resen. Inst. Mat. Estat. Univ. São Paulo 2, No. 3, 293-302 (1996; Zbl 0987.16502)].NEWLINENEWLINEFor the entire collection see [Zbl 0952.00029].