Nilpotent Groups and Their Automorphisms

Description

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021) Notation Preface Part I Linear Methods Chapter 1. Preliminaries § 1.1 Groups § 1.2 Rings and modules § 1.3 Lie rings § 1.4 Mappings, homomorphisms, automorphisms § 1.5 Group actions on a set § 1.6 Fixed points of automorphisms § 1.7 The Jordan normal form of a linear transformation of finite order § 1.8 Varieties and free groups § 1.9 Groups with operators § 1.10 Higman’s Lemma Chapter 2. Nilpotent groups § 2.1 Commutators and commutator subgroups § 2.2 Definitions and basic properties of nilpotent groups § 2.3 Some sufficient conditions for soluble groups to be nilpotent § 2.4 The Schur-Baer Theorem and its converses § 2.5 Lower central series. Isolators § 2.6 Nilpotent groups without torsion § 2.7 Basic commutators and the collecting process § 2.8 Finite p-groups Chapter 3. Associated Lie rings § 3.1 Results on Lie rings analogous to theorems about groups § 3.2 Constructing a Lie ring from a group § 3.3 The Lie ring of a group of prime exponent § 3.4 The nilpotency of soluble Lie rings satisfying the Engel condition Part II Automorphisms Chapter 4. Lie rings admitting automorphisms with few fixed points § 4.1 Extending the ground ring § 4.2 Regular automorphisms of soluble Lie rings § 4.3 Regular automorphisms of Lie rings § 4.4 Almost regular automorphisms of prime order § 4.5 Comments Chapter 5. Nilpotent groups admitting automorphisms of prime order with few fixed points § 5.1 Regular automorphisms of prime order § 5.2 Nilpotent p-groups with automorphisms of order p § 5.3 Nilpotent groups with an almost regular automorphism of prime order § 5.4 Comments Chapter 6. Nilpotency in varieties of groups with operators § 6.1 Preliminary lemmas § 6.2 A nilpotency theorem § 6.3 A local nilpotency theorem § 6.4 Corollaries § 6.5 Comments Chapter 7. Splitting automorphisms of prime order and finite p-groups admitting a partition § 7.1 The connection between splitting automorphisms of prime order and finite p-groups admitting a partition § 7.2 The Restricted Burnside Problem for groups with a splitting automorphism of prime order § 7.3 The structure of finite p-groups admitting a partition and a positive solution of the Hughes problem § 7.4 Bounding the index of the Hughes subgroup § 7.5 Comments Chapter 8. Nilpotent p-groups admitting automorphisms of order pk with few fixed points § 8.1 An application of the Mal’cev correspondence § 8.2 Powerful p-groups § 8.3 A weak bound for the derived length § 8.4 A strong bound for the derived length of a subgroup of bounded index References Index of names Subject Index