Unit groups of group rings. (English) Zbl 0687.16010

Pitman Monographs and Surveys in Pure and Applied Mathematics, 47. Harlow: Longman Scientific & Technical; New York etc.: John Wiley & Sons, Inc. xiii, 393 p. (1989).
The book, though containing a lot of material on group ring units, has appeared at a time at which so much progress regarding the determination of the unit group U of an integral group ring \({\mathbb{Z}}G\) is going on that one cannot but feel a second edition might soon be necessary. The progress is twofold, on the one hand, there has been a break-through with respect to the Zassenhaus conjecture which asks, provided that G is finite, whether a finite subgroup of U is, up to signs, rationally conjugate to a subgroup of G; on the other hand, for a large class of groups, including all nilpotent ones, there have been given explicit generators of a subgroup of finite index in U. Of these latter investigations nothing has been taken into the book; concerning the Zassenhaus conjecture, though, it displays almost our present knowledge. So it contains the known results which have to do with cyclic subgroups of U as well as Weiss’ famous theorem on permutation lattices, which implies the conjecture to be true for p-groups G. The recent proof for nilpotent groups, again by Weiss, is already missing, however; so is the famous counterexample of a special metabelian group given by Roggenkamp and Scott. Moreover, their important results on automorphisms and on the Picard group of \({\mathbb{Z}}G\), which are in close connection with the conjecture, have not been included. Apart from its chapters on the Zassenhaus conjecture, which also contains the present knowledge on the isomorphism problem, the heart of the book is an extensive chapter on units in commutative group rings. This serves as a rich source for results that are known today; it not only covers finite but also infinite groups G. Here are some of its sections: Splitting groups; Conditions for RG to be indecomposable or reduced; Finite generation of U(RG); May’s theorem; Integral domains of characteristic 0; Indecomposable rings; Direct factor theorems; Splitting group algebras; The Ulm-Kaplansky invariants; The torsion free rank; The isomorphism problem. For finite abelian groups the famous results of Bass are represented, which, for groups rings, reflect the theory of cyclotomic units from number theory; again though, a recent discussion of Hoechsmann on the index of the Bass group in the unit group for abelian p-groups has not been entered.
Reviewer: J.Ritter


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