##
**A course in the theory of groups.**
*(English)*
Zbl 0483.20001

Graduate Texts in Mathematics, 80. New York Heidelberg Berlin: Springer- Verlag. xvii, 481 p. DM 98.00; $ 43.60 (1982).

The book under review contains two parts, which however are not strictly distinct. The first 5 chapters present the basic notions and results in group theory. In the following chapters these are completed with new results deepening the previous ones. The book treats finite groups as well as infinite groups, such a treatment being less frequent in the literature. Although intended to be only a course in group theory this book provides to be more, as is stressed in the following analysis.

The first chapter is an introduction to the basic concepts of group theory. Normal closure, core, direct and semi-direct product are also presented. Considering in-finite groups, distinction has been made between restricted and unrestricted direct products. As permutation groups are treated, it is possible to introduce wreath products. The almost forgotten notion of holomorph also occurs. In order to stress the unity of the subject, operator groups and group actions are introduced. The chapter ends with Sylow’s theorems and their applications including Kalulnin’s theorem.

The second chapter deals with classical group theory in a modern fashion. Varieties of groups, verbal and marginal subgroups and connected topics are also presented. In the third chapter are presented different series of groups and, naturally, Jordan-Hölder theorems. As infinite groups are considered, chain conditions are introduced, followed by direct decompositions, Krull-Remak-Schmidt theorems and the study of different cases of simple groups. This chapter ends with finite semisimple groups and the mention of Schreier’s conjecture.

The fourth chapter is an account on abelian groups. The structure theorem of finite abelian groups is given as well as free abelian groups, the structure theorem of abelian groups with minimal condition and Kurosh’s theorems are presented followed by an accurate exposition of Kilikov’s theory. The last paragraph is dedicated to torsion-free groups, Pontryagin’s criterion for freeness and Specker’s theorems on the cartesian sum of infinite cyclic groups.

The fifth chapter treats the classical theory of nilpotent groups. One paragraph is dedicated to p-groups. General results are given, but the main purpose is the study of special types of p-groups. The last paragraph deals with soluble groups and particularly with polycyclic groups and soluble groups with minimal condition.

The sixth chapter deepens the topics from the second one. Nielsen-Schreier, Reidemeister-Schreier, Iwasawa and Magnus theorems are given. Free products are studied and Kurosh’s subgroup theorem is proved. This chapter ends with the study of amalgamated products.

The seventh chapter contains the theory of permutation groups. Sharp transitivity is introduced and studied on different particular groups. The famous Jordan theorem on multiply transitive groups is given completed with the study of Mathieu’s groups. A proof is given of the simplicity of the Mathieu groups.

The eighth chapter is dedicated to the representation theory of groups. The developments of this theory are used to prove some of Burnside’s theorems. Group algebras, character theory, permutation representations are studied. Monomial re-presentations follow together with properties of groups which admit such a representation in algebraically closed fields. The chapter ends with results of Burnside, Frobenius and Wielandt on finite groups.

In the ninth chapter soluble groups are resumed concerning P. Hall’s basic results on Sylow systems, their normalizers and abnormal groups follow. One §is dedicated to \(p\)-soluble and \(p\)-nilpotent groups with stress on groups of \(p\)-length at most 1. Another paragraph deals with supersoluble groups. A brief account on formation theory and Fitting classes is given.

The transfer homomorphism and its applications are exposed in the tenth chapter. Transfers into Sylow subgroups are studied, giving special attention to cyclic Sylow subgroups. One whole paragraph is dedicated to Grün’s theorems and their consequences and another one to Frobenius’s criterion for \(p\)-nilpotency and its applications. The rest of the tenth chapter contains the studies of J. Thompson made with the help of the normalizer of \(J(P)\), where \(J(P)\) is the subgroup generated by all abelian subgroups of \(P\) of maximal rank. A theorem on groups with a nilpotent maximal subgroup is included as well as the fixed point free automorphism theory. The chapter ends with the study of the structure of Frobenius groups.

The eleventh chapter deals with the old problem of group extensions treated in a modern way by means of homological algebra, presented in Gruenberg’s manner. It follows the theory of covering groups, homology and cohomology groups. Standard resolution is deduced from Gruenberg’s resolution. Homology and cohomology groups are interpreted in the general context of group theory insisting on the first three cohomology groups.

In the twelfth chapter nilpotent, soluble and mostly locally nilpotent and locally soluble groups are generalized. Proof is given that the product of two normal locally nilpotent subgroups is locally nilpotent, hence deducing the existence of the Hirsch-Plotkin radical. An account of Malcev and McLain’s research is followed by the study of those locally nilpotent groups that satisfy one of the equivalent properties of finite nilpotent groups which fail to be equivalent in the case of infinite groups. Engel groups and structures are studied within soluble groups and groups with maximal condition. General series are introduced allowing the definition of some generalized soluble groups.

The thirteenth chapter deals with subnormality introduced by Wielandt in 1939, a concept which proved to be of great importance for group theory. The intersection and the join of subnormal subgroups are studied, proving that in the case of finite groups the subnormal subgroups are constituting a lattice while this is not true for the case of infinite groups where the join of two subnormal subgroups is not necessarily a subnormal subgroup. However the join of two subnormal subgroups of an infinite group can be subnormal subgroup in certain cases. A case of this kind is given as an example by the author. A whole paragraph is dedicated to the study of the minimal condition on subnormal subgroups stressing the role of Wielandt’s subgroup defined as the intersection of all normalizers of subnormal subgroups of a group, while another deals with groups where normality is transitive. The chapter ends with the study of automorphism towers and complete groups.

The fourteenth chapter deals with finiteness properties, that is, with the study of infinite groups satisfying certain conditions allowing the deepening of this study. Finitely generated and finitely presented groups are studied by using the homological methods, proving again the high efficiency of these methods. Burnside’s problem is studied indicating the negative answer for the special problem when the exponent is 2,3 or 4 and even some answers in more general cases. One paragraph deals with locally finite groups proving that Sylow subgroups are conjugated only in certain cases. Proof is given for the Hall-Kutilaka-Kargapolov theorem which says that every infinite locally finite group has an infinite abelian subgroup. The chapter ends with the study of 2-groups with maximal or minimal condition and the theory developed by B. H. Neumann about FC-groups.

The last chapter presents researches on infinite soluble groups, soluble linear groups and Malcev’s theory of these groups. It follows the study of soluble groups with finiteness conditions on abelian subgroups. P. Hall’s theory on finitely generated soluble groups, residual finiteness and Frattini subgroups is given. 693 exercises are added to the context of this book. The exercises of each para-graph are a valuable completion of the theory, some of them leading to results used later in the text, some others present important results which have not been incorporated in the text. (Fore instance ex. Nr. 8 and 9 p. 77 dealing with an in-finite simple finitely generated group given by G. Higman.) The book also contains a certain number of interesting and useful results and theorems unproved because of lack of space. (For instance the Felt-Thompson theorem.)

From the above presentation it is clear that the book reviewed is more a little encyclopedia of group theory than a course, being useful to anyone who tries to study group theory. The book has a large bibliography containing the basic books and articles in this field.

Certain few objects can be revealed however. On p. 409 an interesting result of Razmyslov is given without any bibliographical indication. We do not agree with the Pratt G notation of the Frattini subgroup since the \(\Phi(G)\) notation is worldwide accepted and cannot lead to any confusion. On the other hand the notation of semi-direct product is very similar to Huppert’s notation for abnormal subgroups.

As a final conclusion I consider this book as a basic work for anyone who is involved in a thorough study of the theory of groups.

The first chapter is an introduction to the basic concepts of group theory. Normal closure, core, direct and semi-direct product are also presented. Considering in-finite groups, distinction has been made between restricted and unrestricted direct products. As permutation groups are treated, it is possible to introduce wreath products. The almost forgotten notion of holomorph also occurs. In order to stress the unity of the subject, operator groups and group actions are introduced. The chapter ends with Sylow’s theorems and their applications including Kalulnin’s theorem.

The second chapter deals with classical group theory in a modern fashion. Varieties of groups, verbal and marginal subgroups and connected topics are also presented. In the third chapter are presented different series of groups and, naturally, Jordan-Hölder theorems. As infinite groups are considered, chain conditions are introduced, followed by direct decompositions, Krull-Remak-Schmidt theorems and the study of different cases of simple groups. This chapter ends with finite semisimple groups and the mention of Schreier’s conjecture.

The fourth chapter is an account on abelian groups. The structure theorem of finite abelian groups is given as well as free abelian groups, the structure theorem of abelian groups with minimal condition and Kurosh’s theorems are presented followed by an accurate exposition of Kilikov’s theory. The last paragraph is dedicated to torsion-free groups, Pontryagin’s criterion for freeness and Specker’s theorems on the cartesian sum of infinite cyclic groups.

The fifth chapter treats the classical theory of nilpotent groups. One paragraph is dedicated to p-groups. General results are given, but the main purpose is the study of special types of p-groups. The last paragraph deals with soluble groups and particularly with polycyclic groups and soluble groups with minimal condition.

The sixth chapter deepens the topics from the second one. Nielsen-Schreier, Reidemeister-Schreier, Iwasawa and Magnus theorems are given. Free products are studied and Kurosh’s subgroup theorem is proved. This chapter ends with the study of amalgamated products.

The seventh chapter contains the theory of permutation groups. Sharp transitivity is introduced and studied on different particular groups. The famous Jordan theorem on multiply transitive groups is given completed with the study of Mathieu’s groups. A proof is given of the simplicity of the Mathieu groups.

The eighth chapter is dedicated to the representation theory of groups. The developments of this theory are used to prove some of Burnside’s theorems. Group algebras, character theory, permutation representations are studied. Monomial re-presentations follow together with properties of groups which admit such a representation in algebraically closed fields. The chapter ends with results of Burnside, Frobenius and Wielandt on finite groups.

In the ninth chapter soluble groups are resumed concerning P. Hall’s basic results on Sylow systems, their normalizers and abnormal groups follow. One §is dedicated to \(p\)-soluble and \(p\)-nilpotent groups with stress on groups of \(p\)-length at most 1. Another paragraph deals with supersoluble groups. A brief account on formation theory and Fitting classes is given.

The transfer homomorphism and its applications are exposed in the tenth chapter. Transfers into Sylow subgroups are studied, giving special attention to cyclic Sylow subgroups. One whole paragraph is dedicated to Grün’s theorems and their consequences and another one to Frobenius’s criterion for \(p\)-nilpotency and its applications. The rest of the tenth chapter contains the studies of J. Thompson made with the help of the normalizer of \(J(P)\), where \(J(P)\) is the subgroup generated by all abelian subgroups of \(P\) of maximal rank. A theorem on groups with a nilpotent maximal subgroup is included as well as the fixed point free automorphism theory. The chapter ends with the study of the structure of Frobenius groups.

The eleventh chapter deals with the old problem of group extensions treated in a modern way by means of homological algebra, presented in Gruenberg’s manner. It follows the theory of covering groups, homology and cohomology groups. Standard resolution is deduced from Gruenberg’s resolution. Homology and cohomology groups are interpreted in the general context of group theory insisting on the first three cohomology groups.

In the twelfth chapter nilpotent, soluble and mostly locally nilpotent and locally soluble groups are generalized. Proof is given that the product of two normal locally nilpotent subgroups is locally nilpotent, hence deducing the existence of the Hirsch-Plotkin radical. An account of Malcev and McLain’s research is followed by the study of those locally nilpotent groups that satisfy one of the equivalent properties of finite nilpotent groups which fail to be equivalent in the case of infinite groups. Engel groups and structures are studied within soluble groups and groups with maximal condition. General series are introduced allowing the definition of some generalized soluble groups.

The thirteenth chapter deals with subnormality introduced by Wielandt in 1939, a concept which proved to be of great importance for group theory. The intersection and the join of subnormal subgroups are studied, proving that in the case of finite groups the subnormal subgroups are constituting a lattice while this is not true for the case of infinite groups where the join of two subnormal subgroups is not necessarily a subnormal subgroup. However the join of two subnormal subgroups of an infinite group can be subnormal subgroup in certain cases. A case of this kind is given as an example by the author. A whole paragraph is dedicated to the study of the minimal condition on subnormal subgroups stressing the role of Wielandt’s subgroup defined as the intersection of all normalizers of subnormal subgroups of a group, while another deals with groups where normality is transitive. The chapter ends with the study of automorphism towers and complete groups.

The fourteenth chapter deals with finiteness properties, that is, with the study of infinite groups satisfying certain conditions allowing the deepening of this study. Finitely generated and finitely presented groups are studied by using the homological methods, proving again the high efficiency of these methods. Burnside’s problem is studied indicating the negative answer for the special problem when the exponent is 2,3 or 4 and even some answers in more general cases. One paragraph deals with locally finite groups proving that Sylow subgroups are conjugated only in certain cases. Proof is given for the Hall-Kutilaka-Kargapolov theorem which says that every infinite locally finite group has an infinite abelian subgroup. The chapter ends with the study of 2-groups with maximal or minimal condition and the theory developed by B. H. Neumann about FC-groups.

The last chapter presents researches on infinite soluble groups, soluble linear groups and Malcev’s theory of these groups. It follows the study of soluble groups with finiteness conditions on abelian subgroups. P. Hall’s theory on finitely generated soluble groups, residual finiteness and Frattini subgroups is given. 693 exercises are added to the context of this book. The exercises of each para-graph are a valuable completion of the theory, some of them leading to results used later in the text, some others present important results which have not been incorporated in the text. (Fore instance ex. Nr. 8 and 9 p. 77 dealing with an in-finite simple finitely generated group given by G. Higman.) The book also contains a certain number of interesting and useful results and theorems unproved because of lack of space. (For instance the Felt-Thompson theorem.)

From the above presentation it is clear that the book reviewed is more a little encyclopedia of group theory than a course, being useful to anyone who tries to study group theory. The book has a large bibliography containing the basic books and articles in this field.

Certain few objects can be revealed however. On p. 409 an interesting result of Razmyslov is given without any bibliographical indication. We do not agree with the Pratt G notation of the Frattini subgroup since the \(\Phi(G)\) notation is worldwide accepted and cannot lead to any confusion. On the other hand the notation of semi-direct product is very similar to Huppert’s notation for abnormal subgroups.

As a final conclusion I consider this book as a basic work for anyone who is involved in a thorough study of the theory of groups.

Reviewer: Gheorghe Pic (Cluj)