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An introduction to group rings. (English) Zbl 0997.20003

This book presents a specific topic in algebra, namely integral group rings of finite and infinite groups, and is addressed to a reader who is not supposed to be experienced in the subject, in fact, a good knowledge of linear algebra will suffice. Nevertheless, it works its way through to the state of art the subject has reached today.
On its way, it of course needs to provide the means that come from other parts of algebra, especially from group and representation theory, or also from number theory. This is ultimately done parallel to the development of the theory of group rings; the first 200 pages, though, are devoted to collect the facilities that must be known in order to get started: 1. Groups – in particular Abelian, nilpotent and solvable groups, Sylow’s theorem, free groups and products; 2. Rings, Modules and Algebras – in particular Wedderburn’s theorem, algebraic integers and orders, tensor products; 3. Group Rings – in particular semisimplicity and Abelian group algebras; 4. A Glance at Group Representations; 5. Group Characters.
Nothing here is done for the sake of its own; indeed, one never forgets that it is the interest in (integral) group rings which sets the course. Lots of historical remarks accompany the exposition and integrate the topic with general developments in algebra. The deeper results are mostly without proof. However, quite a list of references (181 articles and books) compensates for this unavoidable drawback; the list also contains the newest research papers on group rings. There are occasions, though, where the need for a result and the lack of space for including a proof generate an ambiguous situation in that a reader gets confronted with a specious argument, as for example when the finiteness of the \(\mathbb{Z}\)-rank of the ring of integers in a number field is stated.
With Chapter 6, Ideals in Group Rings, which discusses nilpotent ideals, prime and semiprime group rings as well as chain conditions, the book turns directly to the theory of group rings. Chapter 7 has Zalesskij’s and Kaplansky’s theorem regarding idempotent elements in \(KG\), \(K\) a field and \(G\) any group; moreover, it presents first results on torsion units in \(\mathbb{Z} G\) for finite groups \(G\) and the classification of all finite \(G\) with \(\mathbb{Q} G\) without nilpotent elements. Chapter 8 concentrates on units in group rings: on Bass cyclic and bicyclic units, on group rings having only trivial units, on the presentation of the units in \(\mathbb{Z} G\) for \(G\) finite Abelian and \(G=S_3\), on units in group rings of infinite groups \(G\), on the question when the unit group is finitely generated, and finally on the central units in \(\mathbb{Z} G\) for finite \(G\) and FC-groups \(G\). The discussion about when Bass cyclic and bicyclic units virtually generate the whole unit group of \(\mathbb{Z} G\) for finite \(G\) is omitted.
Chapter 9, The Isomorphism Problem, has the normal subgroup correspondence, circle groups, a short hint at the work of Roggenkamp-Scott, Weiss, and Hertweck, and a longer section on the modular isomorphism problem; dimension subgroups and the Brauer-Jennings-Zassenhaus \(M\)-series show up as well. Chapter 10, Free Groups of Units, gives, among other things, a proof of the freeness of \(\langle u,u^*\rangle\) for a nontrivial bicyclic unit \(u\in\mathbb{Z} G\) and also of a theorem of Hartley and Pickel about the existence of free subgroups in the unit group of \(\mathbb{Z} G\) for finite non-Abelian \(G\). Chapter 11 finally studies properties of the unit group arising from group identities and also discusses when it is solvable or nilpotent.
It is quite conceivable that this book will appeal to students who are looking for a topic that they can specialize in, because it not only clearly outlines what has to be known in order to get in a position to attack concrete open problems but also directly leads to such.

MSC:

20Cxx Representation theory of groups
16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras
20-02 Research exposition (monographs, survey articles) pertaining to group theory
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16S34 Group rings
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16U60 Units, groups of units (associative rings and algebras)
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