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Mathematics 1214: Introduction to Groups, Hilary 2012

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Mathematics 1214: Introduction to Groups, Hilary 2012 Mathematics 1214: introduction to groups, Hilary 2012 Colm OD´ unlaing;´ a few modifications made by Rudolf Tange for 2012 May 26, 2012 Contents 1 Sets and maps 2 2 Set-theoretic operations; functions of several variables 9 3 Binary relations, equivalence relations, and partitions 10 4 Semigroups, monoids, and groups 16 5 Integer division; Zd as an additive group and a multiplicative monoid 18 6 Cardinality: finite sets 21 7 The Cayley table of a finite group 22 8 The symmetric group Sn 23 9 Parity and the alternating group 25 10 Generators for Sn 26 11 Subgroups (and submonoids) 28 11.1 Subgroup generated by a set of elements . 29 12 Matrix groups GL(n); SL(n);O(n); SO(n);U(n); SU(n) 30 13 Symmetry 31 13.1 Symmetries of the regular n-gon; the dihedral group . 32 13.2 Symmetries of the cube . 34 14 Cosets and Lagrange’s Theorem 35 15 Additive subgroups of Z 37 16 Greatest common divisor 38 1 17 Direct products of groups 39 18 Group isomorphism 40 19 Normal subgroups and quotient groups 41 20 Group homomorphisms and the first isomorphism theorem 43 Z× 21 Multiplicative group n , Fermat’s Little Theorem and the Chinese Remainder Theorem 45 22 Group of the cube, again 47 23 Group actions 48 24 A Sylow theorem 50 25 Classification of finite abelian groups 53 25.1 Product of maximal p-groups . 53 25.2 Finite abelian p-group . 54 25.3 Uniqueness . 55 1 Sets and maps (1.1) Mathematical symbols 8 and 9. • 8 is an abbreviation for ‘for all.’ Thus 8x(x + 0 = x) means ‘for all x, x + 0 = x:’ ‘For all,’ ‘for every,’ and ‘for each,’ are pretty well equivalent in mathematics. • A more general usage: (8x 2 R) x2 =6 −1 means ‘for every real number x, x2 =6 −1.’ • 9 is an abbreviation for ‘there exists . such that.’ Thus 8x9y(x + y = 0) means ‘for all x there exists a y such that x + y = 0.’ Equivalently, ‘for some y’ is a better way to read 9y: one can read the above formula as ‘for all x, for some y (x + y = 0).’ • A more general usage: (9x 2 C) x2 = −1 means ‘there exists a complex number x such that x2 = −1.’ 2 (1.2) Cantor’s intuitive definition of a set. A set is a collection of objects considered as a whole. (1.3) Various sets of numbers. The natural numbers N are the nonnegative integers: N = f0; 1; 2;:::g The integers Z are the whole numbers, positive and negative Z = f::: − 3; −2; −1; 0; 1; 2;:::g The rationals Q consist of all fractions p Q = f : p; q 2 Z and q =6 0g q The reals R consist of every number which is the limit of a Cauchy-convergent sequence of rationals (this description must suffice). The complex numbers C are C = fx + iy : x; y 2 Rg where i2 = −1. The quaternions or hypercomplex numbers H are H = fw + ix + jy + kz : w; x; y; z 2 Rg where i2 = j2 = k2 = ijk = −1 (the Broombridge formula). The conjugate of a quaternion q = w + ix + jy + kz is given by q = w − ix − jy − kz and its norm is given by p p jjqjj = qq = w2 + x2 + y2 + z2 : (1.4) Definition A map with domain A and codomain B is a rule or procedure which associates with each x 2 A a unique element of B. The words ‘function, mapping, transformation’ are synonyms for ‘map.’ (1.5) Notation The notation f : A ! B means that A and B are sets and f is a map with domain A and codomain B. Given x 2 A, f(x) denotes the unique element of B associated to x by f. Alternatively, one can write f : x 7! y to mean that y is the unique element of B associated with x by f. (1.6) The identity map ιX , also written idX . If X is any set, there is a well-defined identity map ιX : X ! X; x 7! x for every x 2 X. Frequently the notation idX is used instead of ιX . Other examples. Squaring: R ! R; x 7! x2 is the map which squares each real number x. Parity 3 ( 0 if x is even Z ! Z; x 7! 1 if x is odd More examples of maps. Operations like addition and subtraction can be viewed as certain kinds of map. They are functions of two variables, so at present they don’t fit the pattern, but it will be explained later. Similarly, multiplication, loge (i.e., ln), exponentiation, trigonometric functions, etcetera, can be defined as maps with suitable domains and codomains. (1.7) Restriction of a map. If f : A ! B is a map and A0 ⊆ A, then the restriction of f to A0 is the 0 0 map fjA0 : A ! B defined by fjA0 (x) = f(x) for all x 2 A . (1.8) Equality of maps. • Two maps f : A ! B and g : C ! D are equal if and only if • A = C, • B = D, and • for all x 2 A, f(x) = g(x). They may be given be by completely different formulae. Example. f : N ! Z; x 7! (−1)x g : N ! Z; x 7! 1 − 2x + 4 ∗ (x ÷ 2) Although it is not at all obvious, f = g. (Explicitly, you can show that if x is even then f(x) = g(x) = 1 and if x is odd then f(x) = g(x) = −1. Notice that integer division is used, so if x is even then 4 ∗ (x ÷ 2) = 2x and if x is odd then 4 ∗ (x ÷ 2) = 2x − 2.) (1.9) Range (or image) of a map and image of a set under a map. Given f : A ! B, the range of f is the set ff(x): x 2 Ag It is a subset of B, but not necessarily all of B. More generally, for any subset X of A, we write f(X) = ff(x): x 2 Xg and call f(X) the image of X under f. Thus range(f) = f(A) = f(domain(f)). (1.10) Lemma Given a map f, suppose X and Y are subsets of domain(f). If X ⊆ Y then f(X) ⊆ f(Y ). (Proof: exercise.) Example. Given f : R ! R; x 7! x2 • domain(f) = codomain(f) = R 4 g a f a 1 1 b b 2 2 c c a a 1 1 b b 2 2 c c g f f g Figure 1: f, g, g ◦ f, f ◦ g. • range(f) = [0; 1) • Since N = f0; 1; 2;:::g is (or corresponds to) a subset of R, f(N) = f0; 1; 4; 9 :::g That is, the image of N is (or corresponds to) the set of perfect squares. • Also, Z = f:::−2; −1; 0; 1; 2;:::g is (or corresponds to) a subset of R, and N ⊆ Z. By Lemma 1.10, f(N) ⊆ f(Z). Actually, f(Z) = f(N). (1.11) Compatible maps and composition. Let f : A ! B and g : C ! D be two maps. If range(f) ⊆ domain(g) (i.e., C), then we say g is compatible with f and we define the composite map g ◦ f, g following f, as follows: g ◦ f : A ! D; x 7! g(f(x)): When g is compatible with f we may simply say that the composite map g ◦ f is defined. (1.12) Lemma If domain(g) = codomain(f) then g ◦ f is defined. (Trivial.) (1.13) Lemma (i) range(g ◦ f) = g(range(f)): (ii) Therefore range(g ◦ f) ⊆ range(g). Proof. Let R = range(f). R = ff(x): x 2 domain(f)g g(R) = fg(r): r 2 Rg = fg(f(x)) : x 2 domain(f)g range(g ◦ f) = fg(f(x)) : x 2 domain(f)g ) range(g ◦ f) = g(R); i.e., range(g ◦ f) = g(range(f)): that was (i). 5 R ⊆ domain(g) ) g(R) ⊆ g(domain(g)) (Lemma 1.10); i.e., range(g ◦ f) ⊆ range(g); so (ii) is proved. (1.14) Our definition differs from most (or all) textbooks, including Durbin’s. In Durbin’s book, composition g ◦ f is defined only when f : A ! B and g : B ! D. In other words, the usual practice is to say that g ◦ f is defined iff domain(g) = codomain(f). In our definition, g ◦ f is defined iff domain(g) ⊇ range(f). Now, the ‘codomain’ of a map is rather arbitrary. For example, would you write sin : R ! R; x 7! sin x; or sin : R ! [−1; 1]; x 7! sin x? In the second version the range and codomain are the same, in the first, the range is smaller than the codomain. (1.15) Notice, however, that while the definition of g ◦ f depends on the range (image) of f rather than its codomain, the codomain of g ◦ f is the codomain of g: domain(g ◦ f) = domain(f) and codomain(g ◦ f) = codomain(g): Example. If f(x) = x + 1 and g(x) = x2, with A = B = C = D = R, then g ◦ f(x) = (x + 1)2 and f ◦ g(x) = x2 + 1: Composition of maps is not commutative. This example shows that g ◦ f and f ◦ g need not be equal, even when both are defined. That is, composition of maps is not a commutative operation. However, (1.16) Lemma Composition of maps is associative in the following sense.
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