##
**Finite group theory.**
*(English)*
Zbl 1169.20001

Graduate Studies in Mathematics 92. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4344-4/hbk). xi, 350 p. (2008).

This book reflects the contents of a graduate course, the author offered at the University of Madison at several occasions. As such, it is a gem that every connaisseur and lover of finite group theory should be aware of. The style of presenting it by the author, is at the highest level of didactical presentation and choice of subjects. The keywords in this review deal with the subjects per chapter. Subnormality and transfer are spread out over four chapters.

An unusual feature of the book is, that there are no references to existing literature at all. Hence, as when the interested reader wants to know more on not so generally known theorems or definitions that are mentioned by a name of the inventor (like for instance: Bartels’ theorem, Bochert’s theorem, Cermak-Delgado’s subgroup, and dito theorem, Dietzmann’s theorem, Hartley-Turull’s theorem, Yoshida’s theorem, Venkov’s theorem, etc.), he/she is obliged to do a Google search (or better, to consult the Zentralblatt!) as to titles of papers and year of appearance of them. It is the opinion of the reviewer that this is the only (little?) minus-point of the book. But he hastens to say that we have to do here with an absolute master piece in textbooks on finite group theory. Hence it is highly recommended.

An unusual feature of the book is, that there are no references to existing literature at all. Hence, as when the interested reader wants to know more on not so generally known theorems or definitions that are mentioned by a name of the inventor (like for instance: Bartels’ theorem, Bochert’s theorem, Cermak-Delgado’s subgroup, and dito theorem, Dietzmann’s theorem, Hartley-Turull’s theorem, Yoshida’s theorem, Venkov’s theorem, etc.), he/she is obliged to do a Google search (or better, to consult the Zentralblatt!) as to titles of papers and year of appearance of them. It is the opinion of the reviewer that this is the only (little?) minus-point of the book. But he hastens to say that we have to do here with an absolute master piece in textbooks on finite group theory. Hence it is highly recommended.

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

### MathOverflow Questions:

Operation of a p’-group on a set of p-power order and fix pointsWielandt automorphism tower theorem

### MSC:

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

20B15 | Primitive groups |

20B20 | Multiply transitive finite groups |

20D06 | Simple groups: alternating groups and groups of Lie type |

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

20D15 | Finite nilpotent groups, \(p\)-groups |

20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |

20D25 | Special subgroups (Frattini, Fitting, etc.) |

20D35 | Subnormal subgroups of abstract finite groups |

20D45 | Automorphisms of abstract finite groups |

20E22 | Extensions, wreath products, and other compositions of groups |

20E36 | Automorphisms of infinite groups |