Character theory for the odd order theorem. Transl. from the French by Robert Sandling.

*(English)*Zbl 0940.20001
London Mathematical Society Lecture Note Series. 272. Cambridge: Cambridge University Press. vii, 154 p. (2000).

This book consists of two parts each of which is devoted to the revision of the proof of a central theorem of group theory, proofs where character theory plays a major role.

The first part of the book deals with the famous Feit-Thompson theorem stating that every group of odd order is solvable. One section of the proof is the analysis of the maximal subgroups of a minimal counter example to this theorem. This part has been revised by H. Bender and G. Glauberman [in “Local analysis of the odd order theorem”, Lond. Math. Soc. Lect. Note Ser. 188 (1994; Zbl 0832.20028)]. The odd order paper also contains a character theoretic section (Chap. V) and the concluding section (Chap. VI) where the final contradiction is reached. This last chapter has been revised by the author of the book under review [C. R. Acad. Sci., Paris, Sér. I 299, 531-534 (1984; Zbl 0562.20005)]. There also has been a revision of a substantial part of the character theoretic section by D. A. Sibley [Ill. J. Math. 20, 434-442 (1976; Zbl 0371.20010)]. The author incorporates the result of Sibley and other results of E. C. Dade [Ann. Math., II. Ser. 79, 590-596 (1964; Zbl 0123.25104)] and gives a complete revision of the character theory of the odd order paper. So together with the book of Glauberman and Bender we have now a fully newly written proof of the Feit-Thompson theorem.

Part II of this book is concerned with the revision of M. Suzuki’s so-called \(C\)-groups theorem on doubly transitive groups [Ann. Math., II. Ser. 79, 514-589 (1964; Zbl 0123.25101)] which in turn plays a crucial role in the proof of H. Bender’s strongly embedding theorem [J. Algebra 17, 527-554 (1971; Zbl 0237.20014)]. Based on his article about Suzuki’s theorem [Astérisque 142-143, 141-233, 235-295 (1986; Zbl 0716.20010)] the author gives a simplified proof of this theorem. As in part I of this book again Sibley’s coherence theorem plays an important role.

It is a disturbing fact that large parts of the proofs of results towards the classification of finite simple groups are quite difficult to approach by the nonspecialist. This book is a most valuable contribution to remedy this situation.

The first part of the book deals with the famous Feit-Thompson theorem stating that every group of odd order is solvable. One section of the proof is the analysis of the maximal subgroups of a minimal counter example to this theorem. This part has been revised by H. Bender and G. Glauberman [in “Local analysis of the odd order theorem”, Lond. Math. Soc. Lect. Note Ser. 188 (1994; Zbl 0832.20028)]. The odd order paper also contains a character theoretic section (Chap. V) and the concluding section (Chap. VI) where the final contradiction is reached. This last chapter has been revised by the author of the book under review [C. R. Acad. Sci., Paris, Sér. I 299, 531-534 (1984; Zbl 0562.20005)]. There also has been a revision of a substantial part of the character theoretic section by D. A. Sibley [Ill. J. Math. 20, 434-442 (1976; Zbl 0371.20010)]. The author incorporates the result of Sibley and other results of E. C. Dade [Ann. Math., II. Ser. 79, 590-596 (1964; Zbl 0123.25104)] and gives a complete revision of the character theory of the odd order paper. So together with the book of Glauberman and Bender we have now a fully newly written proof of the Feit-Thompson theorem.

Part II of this book is concerned with the revision of M. Suzuki’s so-called \(C\)-groups theorem on doubly transitive groups [Ann. Math., II. Ser. 79, 514-589 (1964; Zbl 0123.25101)] which in turn plays a crucial role in the proof of H. Bender’s strongly embedding theorem [J. Algebra 17, 527-554 (1971; Zbl 0237.20014)]. Based on his article about Suzuki’s theorem [Astérisque 142-143, 141-233, 235-295 (1986; Zbl 0716.20010)] the author gives a simplified proof of this theorem. As in part I of this book again Sibley’s coherence theorem plays an important role.

It is a disturbing fact that large parts of the proofs of results towards the classification of finite simple groups are quite difficult to approach by the nonspecialist. This book is a most valuable contribution to remedy this situation.

Reviewer: U.Dempwolff (Kaiserslautern)

##### MSC:

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

20C15 | Ordinary representations and characters |

20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |

20B20 | Multiply transitive finite groups |

20D05 | Finite simple groups and their classification |