Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- Chapter 1 Preliminaries
- Chapter 2 Automorphisms and their fixed points
- Chapter 3 Nilpotent and soluble groups
- Chapter 4 Finite p-groups
- Chapter 5 Lie rings
- Chapter 6 Associated Lie rings
- Chapter 7 Regular automorphisms of Lie rings
- Chapter 8 Almost regular automorphism of order p: almost nilpotency of p-bounded class
- Chapter 9 The Baker–Hausdorff Formula and nilpotent ℚ-powered groups
- Chapter 10 The correspondences of A. I. Mal'cev and M. Lazard
- Chapter 11 Powerful p-groups
- Chapter 12 Almost regular automorphism of order pn: almost solubility of pn-bounded derived length
- Chapter 13 p-Automorphisms with p fixed points
- Chapter 14 Automorphism of order p with pm fixed points: almost nilpotency of m-bounded class
- Bibliography
- Index of names
- Subject Index
- List of symbols
Chapter 1 - Preliminaries
Published online by Cambridge University Press: 22 October 2009
- Frontmatter
- Contents
- Preface
- Introduction
- Chapter 1 Preliminaries
- Chapter 2 Automorphisms and their fixed points
- Chapter 3 Nilpotent and soluble groups
- Chapter 4 Finite p-groups
- Chapter 5 Lie rings
- Chapter 6 Associated Lie rings
- Chapter 7 Regular automorphisms of Lie rings
- Chapter 8 Almost regular automorphism of order p: almost nilpotency of p-bounded class
- Chapter 9 The Baker–Hausdorff Formula and nilpotent ℚ-powered groups
- Chapter 10 The correspondences of A. I. Mal'cev and M. Lazard
- Chapter 11 Powerful p-groups
- Chapter 12 Almost regular automorphism of order pn: almost solubility of pn-bounded derived length
- Chapter 13 p-Automorphisms with p fixed points
- Chapter 14 Automorphism of order p with pm fixed points: almost nilpotency of m-bounded class
- Bibliography
- Index of names
- Subject Index
- List of symbols
Summary
Here we record some basic definitions and elementary results about group, rings, and varieties of algebraic systems. We assume a basic knowledge in undergraduate algebra. In particular, in group theory, the reader is supposed to be familiar with definitions and basic properties of subgroups, cosets, cyclic subgroups, direct products, the structure of finite abelian groups, normal subgroups, the Homomorphism Theorems. The Sylow Theorems may be referred to occasionally, but they are not used in the proofs of the main results of the book, which are all about finite p-groups. Some familiarity with rings and modules is assumed, although many of the definitions are briefly reproduced.
We shall often use exponent notation for images under mappings, that is, aφ or Aφ for φ(a) or φ(A) respectively, and sometimes also the right operator notation, aφ for φ(a), say. The identity mapping of a set M will be denoted by 1M. Standard notation ℕ, ℤ, ℚ, ℝ, ℂ is fixed for the sets of natural numbers, integers, rational, real and complex numbers, respectively. We shall say that a value is (a, b)-bounded, say, if there is a function depending only on a and b, such that the value does not exceed this function.
Groups
Some basic definitions. The sets ℤ, ℚ, ℝ, ℂ are groups with respect to addition. The set ℤ/nℤ of residues modulo n is a group with respect to addition mod n. Every vector space is a group with respect to addition.
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- Information
- p-Automorphisms of Finite p-Groups , pp. 1 - 25Publisher: Cambridge University PressPrint publication year: 1998