Units in integral group rings.

*(English)*Zbl 0803.16022
Pitman Monographs and Surveys in Pure and Applied Mathematics. 69. Harlow: Longman Scientific & Technical. New York, N.Y.: John Wiley & Sons, xiii, 357 p. (1993).

This is, in some sense, a continuation of the author’s earlier book Topics in Group Rings [Marcel Dekker, New York (1978; Zbl 0411.16004)]. While the new book is independent of the older one, it is both interesting and gratifying to compare the two, to see how the subject has developed and matured over the past 15 years. The present book is a well- written account of several new and exciting results on units in integral group rings.

Let \(R\) be a ring, let \(G\) be a multiplicative group, and let \(RG\) denote the group ring of \(G\) over \(R\). If \(R = \mathbb{Z}\) is the ring of rational integers, then the famous isomorphism question asks whether the group ring \(\mathbb{Z} G\) determines the isomorphism class of \(G\). For the most part, this question concerns finite groups, and for the most part, the book under review studies finite groups. Since \(G\) embeds naturally in the unit group \({\mathcal U}(\mathbb{Z} G)\) of \(\mathbb{Z} G\), and in fact in the units \({\mathcal U}_ 1(\mathbb{Z} G)\) of augmentation 1, it is natural to attack the problem by studying units in group rings. This is the theme of the book.

Chapter 1, Introductory Results, contains the basic definitions and some useful techniques. It includes the theorem of Higman characterizing those finite groups \(G\) with \({\mathcal U}(\mathbb{Z} G) = \pm G\), and the result of Hartley-Pickel which determines when \({\mathcal U}(\mathbb{Z} G)\) contains a free subgroup of rank 2. In addition \({\mathcal U}(\mathbb{Z} S_ 3)\) and \({\mathcal U}_ 1(\mathbb{Z} D_ 8)\) are computed in detail.

The goal of the next two chapters is to obtain generators for large subgroups of \({\mathcal U}_ 1(\mathbb{Z} G)\). Chapter 2, Abelian Group Rings, handles the abelian case. It starts by introducing Hoechsmann units, alternating units, and more importantly the Bass cyclic units. It then obtains the theorem of Bass which asserts that if \(C\) is cyclic of order \(> 3\), then the Bass cyclic units generate a torsion-free subgroup of finite index in \({\mathcal U}(\mathbb{Z} C)\). This is followed by the result of Bass-Milnor which extends the preceding theorem to arbitrary finite abelian groups \(A\). In addition, some information is offered on the index in \({\mathcal U}(\mathbb{Z} A)\) of the subgroup generated by the Bass cyclic units of \(\mathbb{Z} C\) as \(C\) runs through the cyclic subgroups of \(A\).

Chapter 3, Units in Noncommutative Group Rings, studies the nonabelian case, and introduces the bicyclic units. If \({\mathcal B}_ 1\) denotes the set of Bass cyclic units and \({\mathcal B}_ 2\) is the set of bicyclics, then the question of interest is whether \(\langle {\mathcal B}_ 1,{\mathcal B}_ 2\rangle\) has finite index in \({\mathcal U}(\mathbb{Z} G)\). The answer here is not always affirmative. The results in this chapter depend on rather deep properties of \(SL_ n(\mathbb{Z})\), and a key conclusion is the theorem of Kleinert which offers a sufficient condition for the normal closure of \(\langle{\mathcal B}_ 1\rangle\) to have finite index in \({\mathcal U}(\mathbb{Z} G)\).

Chapter 4, Normal Complements, considers whether the group \(G\) has a normal complement in \({\mathcal U}_ 1(\mathbb{Z} G)\) and in particular, whether this complement is torsion free. An important positive result here, due to Cliff-Sehgal-Weiss, handles most metabelian groups. Even so, the chapter ends with a corollary of Sekiguchi which asserts that \(G = C_ p\rtimes C_ 8\) (with the obvious action when \(p \equiv 1 \bmod 8\)) has a normal complement in \({\mathcal U}_ 1(\mathbb{Z} G)\) if and only if \(p = 17\).

Chapter 5, Zassenhaus Conjectures, concerns a number of strong versions of the isomorphism question. Specifically, (ZC3) asks whether any finite subgroup \(H\) of \({\mathcal U}_ 1(\mathbb{Z} G)\) is necessarily conjugate to a subgroup of \(G\) by a unit in the rational group algebra \(\mathbb{Q} G\), while (ZC2) asks the same question but under the assumption that \(| H| = | G|\). In the case of nilpotent groups, (ZC2) was shown to be true by Roggenkamp-Scott, and (ZC3) was proved by Weiss. This chapter contains the complete argument of Weiss and discusses the counterexample of Roggenkamp-Scott to (ZC2) with \(G\) a particular metabelian group.

Chapter 6, Infinite Groups, considers \({\mathcal U}(\mathbb{Z} G)\) when \(G\) is not necessarily finite. Here the results are “rudimentary”. For example, there is a conjecture that if \(G\) is torsion free and \(K\) is any field, then \({\mathcal U}KG = K^* G\) consists of trivial units. This appears to be considerably more difficult than the zero divisor conjecture, where at least some positive results are known. But essentially nothing positive is known on the unit conjecture. In a more concrete direction, the chapter contains results of Levin-Sehgal and Wallace on the infinite dihedral group.

Chapter 7, Research Problems, lists and briefly discusses 56 open problems of interest. Finally, there is an Appendix, Rigidity of \(\pi\)- adic \(p\)-torsion, written by Al Weiss, which contains material related to his work on the Zassenhaus conjecture (ZC3).

Let \(R\) be a ring, let \(G\) be a multiplicative group, and let \(RG\) denote the group ring of \(G\) over \(R\). If \(R = \mathbb{Z}\) is the ring of rational integers, then the famous isomorphism question asks whether the group ring \(\mathbb{Z} G\) determines the isomorphism class of \(G\). For the most part, this question concerns finite groups, and for the most part, the book under review studies finite groups. Since \(G\) embeds naturally in the unit group \({\mathcal U}(\mathbb{Z} G)\) of \(\mathbb{Z} G\), and in fact in the units \({\mathcal U}_ 1(\mathbb{Z} G)\) of augmentation 1, it is natural to attack the problem by studying units in group rings. This is the theme of the book.

Chapter 1, Introductory Results, contains the basic definitions and some useful techniques. It includes the theorem of Higman characterizing those finite groups \(G\) with \({\mathcal U}(\mathbb{Z} G) = \pm G\), and the result of Hartley-Pickel which determines when \({\mathcal U}(\mathbb{Z} G)\) contains a free subgroup of rank 2. In addition \({\mathcal U}(\mathbb{Z} S_ 3)\) and \({\mathcal U}_ 1(\mathbb{Z} D_ 8)\) are computed in detail.

The goal of the next two chapters is to obtain generators for large subgroups of \({\mathcal U}_ 1(\mathbb{Z} G)\). Chapter 2, Abelian Group Rings, handles the abelian case. It starts by introducing Hoechsmann units, alternating units, and more importantly the Bass cyclic units. It then obtains the theorem of Bass which asserts that if \(C\) is cyclic of order \(> 3\), then the Bass cyclic units generate a torsion-free subgroup of finite index in \({\mathcal U}(\mathbb{Z} C)\). This is followed by the result of Bass-Milnor which extends the preceding theorem to arbitrary finite abelian groups \(A\). In addition, some information is offered on the index in \({\mathcal U}(\mathbb{Z} A)\) of the subgroup generated by the Bass cyclic units of \(\mathbb{Z} C\) as \(C\) runs through the cyclic subgroups of \(A\).

Chapter 3, Units in Noncommutative Group Rings, studies the nonabelian case, and introduces the bicyclic units. If \({\mathcal B}_ 1\) denotes the set of Bass cyclic units and \({\mathcal B}_ 2\) is the set of bicyclics, then the question of interest is whether \(\langle {\mathcal B}_ 1,{\mathcal B}_ 2\rangle\) has finite index in \({\mathcal U}(\mathbb{Z} G)\). The answer here is not always affirmative. The results in this chapter depend on rather deep properties of \(SL_ n(\mathbb{Z})\), and a key conclusion is the theorem of Kleinert which offers a sufficient condition for the normal closure of \(\langle{\mathcal B}_ 1\rangle\) to have finite index in \({\mathcal U}(\mathbb{Z} G)\).

Chapter 4, Normal Complements, considers whether the group \(G\) has a normal complement in \({\mathcal U}_ 1(\mathbb{Z} G)\) and in particular, whether this complement is torsion free. An important positive result here, due to Cliff-Sehgal-Weiss, handles most metabelian groups. Even so, the chapter ends with a corollary of Sekiguchi which asserts that \(G = C_ p\rtimes C_ 8\) (with the obvious action when \(p \equiv 1 \bmod 8\)) has a normal complement in \({\mathcal U}_ 1(\mathbb{Z} G)\) if and only if \(p = 17\).

Chapter 5, Zassenhaus Conjectures, concerns a number of strong versions of the isomorphism question. Specifically, (ZC3) asks whether any finite subgroup \(H\) of \({\mathcal U}_ 1(\mathbb{Z} G)\) is necessarily conjugate to a subgroup of \(G\) by a unit in the rational group algebra \(\mathbb{Q} G\), while (ZC2) asks the same question but under the assumption that \(| H| = | G|\). In the case of nilpotent groups, (ZC2) was shown to be true by Roggenkamp-Scott, and (ZC3) was proved by Weiss. This chapter contains the complete argument of Weiss and discusses the counterexample of Roggenkamp-Scott to (ZC2) with \(G\) a particular metabelian group.

Chapter 6, Infinite Groups, considers \({\mathcal U}(\mathbb{Z} G)\) when \(G\) is not necessarily finite. Here the results are “rudimentary”. For example, there is a conjecture that if \(G\) is torsion free and \(K\) is any field, then \({\mathcal U}KG = K^* G\) consists of trivial units. This appears to be considerably more difficult than the zero divisor conjecture, where at least some positive results are known. But essentially nothing positive is known on the unit conjecture. In a more concrete direction, the chapter contains results of Levin-Sehgal and Wallace on the infinite dihedral group.

Chapter 7, Research Problems, lists and briefly discusses 56 open problems of interest. Finally, there is an Appendix, Rigidity of \(\pi\)- adic \(p\)-torsion, written by Al Weiss, which contains material related to his work on the Zassenhaus conjecture (ZC3).

Reviewer: D.S.Passman (Madison)

##### MSC:

16S34 | Group rings |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20E07 | Subgroup theorems; subgroup growth |

16U60 | Units, groups of units (associative rings and algebras) |

20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |

20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |

20D40 | Products of subgroups of abstract finite groups |