Notation and Conventions
These will be in effect throughout the book and include the following: Let X be a set. If x is an element of X, we will write x∈X, and we will at times denote the subset {x} of X by x also. The cardinality (either finite or infinite) of X is denoted card
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These will be in effect throughout the book and include the following: Let X be a set. If x is an element of X, we will write XEX, and we will at times denote the subset {x} of X by x also. The cardinality (either finite or infinite) of X is denoted card X and on occasion by IXI ifit is finite. The Cartesian product of two sets X and Y is denoted X x Y. The identity map of X is denoted id x. For subsets Yand Z of X, Z ~ Yor Y;2 Z denote inclusion, Z c Yor Y:::J Z denote strict inclusion, and X - Y denotes set theoretic difference. If Y is a subset of X, then Y ~ X is the inclusion map. Maps will ordinarily be written on the left. In particular the composite fog of f and 9 will mean that one first applies 9 and then f. Let f:X --+Z be a map. The imagef(X) of X under f will often be written imf. The restriction of f to a subset Y of X is denoted by fly, or when less precision is required, by f: Y --+ Z. Let I be another set. Suppose that to each iEI, there corresponds an element Xi' i.e. suppose that we have a function f: I --+ X. The subset f(1) of X will on occasion be denoted {XJiEI' and we will say that it is indexed by I. At times I will be linearly or totally ordered, i.e. it will have a partial ordering ~ such that for any i and j in I, either i ~ j or j ~ i. Let X be a group. The product in X is generally denoted by juxtaposition of elements. On occasion"·" is used for emphasis. If X is an additive group, LfinX denotes an arbitrary sum of finitely many elements of X. Let X and Y be groups. The kernel of a homomorphism f: X --+ Y is denoted kerf. An isomorphism from X to Y is understood to be bijective. If X and Yare isomorphic we write X ~ Y. An isomorphism f:X --+ Y will also be written f:X ~ Y to emphasize the bijectivity of f. The group of automorphisms of X is denoted Aut X. The Cartesian product X x Y becomes a group by using the operation that is given componentwise. Analogous terminology and notation will be in effect for rings, modules, etc. The ring of integers is denoted by 7L and the set of natural numbers {kE7Llk > O} by N. The rational, real, and complex numbers are denoted i1J, IR, and C respectively. The greatest common divisor of two natural numbers k and n is written gcd(k, n). For any x in IR, [x] denotes the largest integer less than or equal to x. If d = pk is a prime power, then there is up to isomorphism a unique field with d elements. We will denote this Galois field by IF d' This book consists of nine chapters each of which is subdivided into a number of "sections", and these again are partitioned into "paragraphs" labelled "§". For example Section 1.5 of Chapter 1 consists of §1.5A and §1.5B.
A. J. Hahn et al., The Classical Groups and K-Theory © Springer-Verlag Berlin Heidelberg 1989
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Notation and Conventions
The following "*,, convention will be in effect throughout: The sections, paragraphs, theorems, and propositions where proofs, constructions, etc., are not given in detail but are only sketched (and also those which logically depend on such material) are designated with a
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