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**Characters of finite groups. Part 2. Transl. from the orig. Russian manuscript by Ya. Berkovich and V. Zobina. Transl. edited by David Louvish.**
*(English)*
Zbl 0934.20009

Translations of Mathematical Monographs. 181. Providence, RI: American Mathematical Society (AMS). xxv, 332 p. (1998).

This review covers also volume I, see the preceding item Zbl 0934.20008.

Also the translation into English from the Russian original took more than about 11 years, accompanied by its publication, this does not mean that the book contains results up to 1985 or so. On the contrary, it is meant that the contents of the book are not only self-contained, but they have also been brought up to date.

There is a lot to learn and to read here, for students, freshmen and specialists. The authors themselves start with a Preface of seven pages long, in which they explain their intentions: the reader of this review can do no better than to consult that preface. Let us take a look at the contents. There are 31 chapters. Almost all contain exercises, informative remarks, appendices and notes. There is an immense bibliography in both volumes.

The following is an overview of the contents without being complete. 1. Basic concepts: modules, algebras, representations, Schur’s lemma, Maschke’s theorem. 2. Characters: orthogonality relations, Theorems of Nagao, Gallagher, Saksonov. 3. Arithmetical properties of characters: Theorem of Frobenius and Molien, conjugate characters, rational groups, class equation. 4. Products of characters: tensor product, exterior and symmetric powers, Frobenius-Schur-indicator, Mann’s theorem. 5. Induced characters and representations: Mackey’s theorems, Theorems of Brauer-Suzuki-Wall, monomial representations 6. Projective representations: Schur multiplier, Reynold’s realization theorem, \(p\)-groups with large multipliers. 7. Clifford theory: Theorems of Ito, Clifford, Gallagher, Isaacs, Dornhoff, Tate, Dade. 8. Brauer’s induction theorems. 9. Faithful representations: Theorems of Zhmud’ and Weisner. 10. Existence of normal subgroups: Frobenius groups, Wielandt-Frobenius theorems. 12. Sums of degrees of irreducible characters. 12. Groups of relatively small height. 13. The Brauer-Suzuki theorem. 14. Degrees and kernels of irreducible characters: Theorems of Garrison, Blichfeldt, Isaacs and Passman, Thompson, Chillag and Herzog, Minkowski, Taketa. 15. Involutions: real elements, Bender’s method, Brauer’s and Thompson’s formulas. 16. Connectedness and Zassenhaus groups. 17. Theorem of Nagao. 18. Linear groups. 19. Permutation characters: orbits, characters of \(n\)-transitive permutation groups, characters of \(S_5\) and \(S_6\), Young theory. 20. Characters of \(\text{SL}(2,p^n)\). 21. Zeroes of characters. 22. The Schur index. 23. Degrees of irreducible components of induced characters. In chapters 24 up to and including 27 properties of character degrees are shown, such as counting characters of some specified degree properties. 28. Nonsolvable groups with many involutions. 29. Kernels of nonlinear irreducible characters. 30. Monolithic characters. 31. The class number: Landau’s theorem, algebraically conjugate characters, groups with at most three nonlinear irrducible characters, \(p\)-groups with few nonlinear irreducible characters.

There is a “problem” section of 15 pages added; it consists of unsolved problems and questions, although the reviewer wishes to remark that due to time-elaps of the publication of the book, some of them have been solved now.

The imprint and lay-out of both volumes is excellent. The reviewer regards the book as a “must”, certainly due to the techniques and results offered, it is up to date, and it is (together with I. M. Isaacs’ book: Character theory of finite groups (1976; Zbl 0337.20005) and B. Huppert’s book under that very same title (1998; Zbl 0932.20007)) a very welcome and illuminating contribution in presenting to the mathematical community, the beauty of the theory of characters of finite groups.

Also the translation into English from the Russian original took more than about 11 years, accompanied by its publication, this does not mean that the book contains results up to 1985 or so. On the contrary, it is meant that the contents of the book are not only self-contained, but they have also been brought up to date.

There is a lot to learn and to read here, for students, freshmen and specialists. The authors themselves start with a Preface of seven pages long, in which they explain their intentions: the reader of this review can do no better than to consult that preface. Let us take a look at the contents. There are 31 chapters. Almost all contain exercises, informative remarks, appendices and notes. There is an immense bibliography in both volumes.

The following is an overview of the contents without being complete. 1. Basic concepts: modules, algebras, representations, Schur’s lemma, Maschke’s theorem. 2. Characters: orthogonality relations, Theorems of Nagao, Gallagher, Saksonov. 3. Arithmetical properties of characters: Theorem of Frobenius and Molien, conjugate characters, rational groups, class equation. 4. Products of characters: tensor product, exterior and symmetric powers, Frobenius-Schur-indicator, Mann’s theorem. 5. Induced characters and representations: Mackey’s theorems, Theorems of Brauer-Suzuki-Wall, monomial representations 6. Projective representations: Schur multiplier, Reynold’s realization theorem, \(p\)-groups with large multipliers. 7. Clifford theory: Theorems of Ito, Clifford, Gallagher, Isaacs, Dornhoff, Tate, Dade. 8. Brauer’s induction theorems. 9. Faithful representations: Theorems of Zhmud’ and Weisner. 10. Existence of normal subgroups: Frobenius groups, Wielandt-Frobenius theorems. 12. Sums of degrees of irreducible characters. 12. Groups of relatively small height. 13. The Brauer-Suzuki theorem. 14. Degrees and kernels of irreducible characters: Theorems of Garrison, Blichfeldt, Isaacs and Passman, Thompson, Chillag and Herzog, Minkowski, Taketa. 15. Involutions: real elements, Bender’s method, Brauer’s and Thompson’s formulas. 16. Connectedness and Zassenhaus groups. 17. Theorem of Nagao. 18. Linear groups. 19. Permutation characters: orbits, characters of \(n\)-transitive permutation groups, characters of \(S_5\) and \(S_6\), Young theory. 20. Characters of \(\text{SL}(2,p^n)\). 21. Zeroes of characters. 22. The Schur index. 23. Degrees of irreducible components of induced characters. In chapters 24 up to and including 27 properties of character degrees are shown, such as counting characters of some specified degree properties. 28. Nonsolvable groups with many involutions. 29. Kernels of nonlinear irreducible characters. 30. Monolithic characters. 31. The class number: Landau’s theorem, algebraically conjugate characters, groups with at most three nonlinear irrducible characters, \(p\)-groups with few nonlinear irreducible characters.

There is a “problem” section of 15 pages added; it consists of unsolved problems and questions, although the reviewer wishes to remark that due to time-elaps of the publication of the book, some of them have been solved now.

The imprint and lay-out of both volumes is excellent. The reviewer regards the book as a “must”, certainly due to the techniques and results offered, it is up to date, and it is (together with I. M. Isaacs’ book: Character theory of finite groups (1976; Zbl 0337.20005) and B. Huppert’s book under that very same title (1998; Zbl 0932.20007)) a very welcome and illuminating contribution in presenting to the mathematical community, the beauty of the theory of characters of finite groups.

Reviewer: R.W.van der Waall (Amsterdam)

### MSC:

20C15 | Ordinary representations and characters |

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

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |