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The multiplicative group of an integral group ring. (Мультипликативная группа целочисленного группового кольца.) (Russian) Zbl 0688.16007
Uzhgorod: Uzhgorodskiĭ Gosudarstvennyĭ Universitet. 210 p. (1987).
Group algebras came into scene in the papers of Cayley, Frobenius, Molien and Schur and were primarily intended to give a better understanding of finite group representations. In the 1950s group algebras for infinite groups also became a frequent object of study. Nowadays group rings can be viewed as an independent branch of Algebra with a number of deep results, sophisticated techniques and with a lot of applications. The structure of the group plays a central role in these things. However, problems about Group Rings take their form with the help of Ring Theory. And last but not least, group rings act also as representatives of rings of ‘graduate type’. There exists a very vast literature on this topic. Let’s only point out two books [D. Passman, The algebraic structure of group rings. New York etc.: John Wiley & Sons (1977; Zbl 0368.16003)], S. Sehgal, Topics in group rings. New York etc. : Marcel Dekker (1978; Zbl 0411.16004)] and a survey [A. Zalesskii, A. Mikhal’ev, Group Rings, in Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 2, 5–118 (1973; Zbl 0293.16013)].
These books have in common several basic results about the multiplicative group $$U(K[G])$$ of a group ring. G. Higman [Proc. Lond. Math. Soc., II. Ser. 46, 231–248 (1940; Zbl 0025.24302)] was the first to find deep results about $$U(\mathbb{Z}[G])$$, comments on Higman’s dissertation see in [R. Sandling, Lect. Notes Math. 882, 93–116 (1981; Zbl 0468.16013)]. In Sehgal’s book a successful attempt has been undertaken to describe systematically the results about $$U(K[G])$$ obtained so far. The author’s book gives another systematic treatment of $$U(\mathbb{Z}[G])$$ where many recent results (including those of the author and his students) are considered. It contains 4 chapters: (1) Preliminaries; (2) Structure of $$U(\mathbb{Z}[G])$$; (3) The unitary subgroup of $$U(\mathbb{Z}[G])$$; (4) Subgroups of $$U(\mathbb{Z}[G])$$.
The readers are supposed to have some acquaintance with groups, rings and algebraic numbers. They will find here a systematic and in-depth coverage with full proofs of many questions about $$U(\mathbb{Z}[G])$$ considered so far in papers only.
Let’s give some details of the contents of this text. Chapter 1 begins with an introduction to group rings. Connections with crossed products are considered, this last notion being commented also. Some classics about units in rings of algebraic integers follow.
Chapter 2 contains the main bulk of results about $$U(\mathbb{Z}[G])$$. The subgroup of normalized units $$V(\mathbb{Z}[G])=\{x\in U(\mathbb{Z}[G])\mid\sum_{g\in G}x_g=1\}$$ in $$U(\mathbb{Z}[G])$$ is introduced and a number of results about $$V(\mathbb{Z}[G])$$ are proved. These results are based on a generalization to a larger class of (commutative) rings $$K$$ instead of $$\mathbb{Z}$$ of a result of G. Higman and S. Berman: the trace of a nontrivial element of finite order in $$V(\mathbb{Z}[G])$$ for a finite group $$G$$ equals 0. Then it follows (§3) a description (with the help of invariants) of $$V(\mathbb{Z}[G])$$ for a finite abelian group $$G$$; e.g. the results of R. Ayoub and Chr. Ayoub [Bull. Aust. Math. Soc. 1, 245–261 (1969; Zbl 0172.31403)] are presented here with proofs. Then free generators are found for a free Abelian subgroup of $$V(\mathbb{Z}[G])$$ for a finite cyclic group and theorems of H. Bass and J. Milnor are proved. In §5 some theorems concerning the conjecture ‘For any torsion-free group $$G$$ the group $$V(\mathbb{Z}[G])$$ is trivial’ are presented; e.g. some results about the equality $$V(\mathbb{Z}[G])=G\cdot V(\mathbb{Z}[\pi (G)])$$ are considered here. §6 investigates the structure of unit groups of group rings for circle groups, i.e. for those groups $$G$$ which have the form $$(R,\circ)$$ for some radical ring $$R$$; here $$a\circ b=a+b-ab$$, $$a,b\in R$$. Known results of L. Kruse, J. Ault and J. Watters, D. Passman and P. Smith and the author are given and a number of consequences about $$V(\mathbb{Z}[G])$$ being nilpotent of degree $$\leq 2$$ are derived.
S. Berman gave (1955) a description of finite groups G with V($${\mathbb{Z}}[G])$$ having its elements of finite order trivial. Results of A. Bovdi and S. Sehgal concerning an analogous problem for nonperiodic group G are given in §7. §8 considers the question ‘when it happens that the centres of G and V($${\mathbb{Z}}[G])$$ coincide’. §§9-10 contain, in the reviewer’s opinion, some main results of this book. The author shows that investigation of various properties (nilpotency, local nilpotency, FC-property, etc.) of U($${\mathbb{Z}}[G])$$ essentially takes to the question of describing $${\mathbb{Z}}[G]$$ with U($${\mathbb{Z}}[G])$$ having some periodic invariant subgroup. This last question was considered firstly by S. Berman and A. Rossa (1966). Here some sophisticated results of the author [Sib. Mat. Zh. 11, 492–511 (1970; Zbl 0216.078)] are presented about $${\mathbb{Z}}[G]$$ with V($${\mathbb{Z}}[G])$$ having all its elements of finite order trivial. Thereafter, §11 is devoted to the case where there exist nontrivial units of finite order in U($${\mathbb{Z}}[G])$$. This § begins with D. Passman and P. Smith’s results [J. Algebra 69, 213–239 (1981; Zbl 0459.16008)] and gives then a survey of results of I. Hughes and K. Pearson (1972) and J. Ritter and S. Sehgal (1982) also among other things.
Chapter 3 is devoted to the author’s results on the following question. For an element $$x=\sum_{g\in G}x_gg$$ in $$\mathbb{Z}[G]$$ let’s denote $$x^ f=\sum_{g\in G}x_ gf(g)g^{-1}$$ for a homomorphism $$f: G\to U(\mathbb{Z})$$. All elements $$u$$ in $$U(\mathbb{Z}[G])$$ with the property $$u^{-1}\in \{u^ f,-u^f\}$$ give us the $$f$$-unitary subgroup $$U_f(\mathbb{Z}[G])$$. The question (S. Novikov) is to describe the structure of this subgroup $$U_f(\mathbb{Z}[G])$$. The author’s results [Mat. Sb., Nov. Ser. 119, No.3, 387–400 (1982; Zbl 0511.16009)] are presented: some necessary and some sufficient conditions for $$U_f(\mathbb{Z}[G]) = U(\mathbb{Z}[G])$$ are given. The present form of these conditions is, perhaps, too lengthy and so they seem somewhat artificial. However, they are, perhaps, useful in topology and the author hopes them to be the core for giving to this topic a more finished form.
The last chapter 4 is about the subgroup structure of $$U(\mathbb{Z}[G])$$. After some preparatory work the main theorem from B. Hartley and P. Pickel’s paper [Can. J. Math. 32, 1342–1352 (1980; Zbl 0458.16007)] is given in §15. Then some properties of $$U(\mathbb{Z}[G])$$ it shares with free groups, are considered. The question about the existence of a normal complement for $$G$$ in $$V(\mathbb{Z}[G])$$ is investigated and some results of G. Cliff, S. Sehgal and A. Weiss [J. Algebra 73, 167–185 (1981; Zbl 0484.16004)] are presented in §17. The last §18 is about conjugacy of elements of finite order and about finite subgroups in $$V(K[G])$$; some recent theorems of A. Bovdi and M. Dokuchaev are presented, also K. Roggenkamp and L. Scott’s recent results are commented here.

MSC:
 16U60 Units, groups of units (associative rings and algebras) 16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras 16S34 Group rings 20-02 Research exposition (monographs, survey articles) pertaining to group theory 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 20E07 Subgroup theorems; subgroup growth 20C05 Group rings of finite groups and their modules (group-theoretic aspects)