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1 Monoids and Groups 1 Monoids and groups 1.1 Definition. A monoid is a set M together with a map M × M ! M; (x; y) 7! x · y such that (i) (x · y) · z = x · (y · z) 8x; y; z 2 M (associativity); (ii) 9e 2 M such that x · e = e · x = x for all x 2 M (e = the identity element of M). 1.2 Examples. 1) Z with addition of integers (e = 0) 2) Z with multiplication of integers (e = 1) 3) Mn(R) = fthe set of all n × n matrices with coefficients in Rg with ma- trix multiplication (e = I = the identity matrix) 4) U = any set P (U) := fthe set of all subsets of Ug P (U) is a monoid with A · B := A [ B and e = ?. 5) Let U = any set F (U) := fthe set of all functions f : U ! Ug F (U) is a monoid with multiplication given by composition of functions (e = idU = the identity function). 1.3 Definition. A monoid is commutative if x · y = y · x for all x; y 2 M. 1.4 Example. Monoids 1), 2), 4) in 1.2 are commutative; 3), 5) are not. 1 1.5 Note. Associativity implies that for x1; : : : ; xk 2 M the expression x1 · x2 ····· xk has the same value regardless how we place parentheses within it; e.g.: (x1 · x2) · (x3 · x4) = ((x1 · x2) · x3) · x4 = x1 · ((x2 · x3) · x4) etc. 1.6 Note. A monoid has only one identity element: if e; e0 2 M are identity elements then e = e · e0 = e0 1.7 Definition. A group is a monoid G such that for any x 2 G there is y 2 G satistying x · y = e = y · x. The element y is called the inverse of x and it is denoted by x−1 (or by −x in the additive notation). A group G is commutative (or abelian) if x · y = y · x for all x; y 2 G. 1.8 Examples. 1) Z; Q; R; C with addition 2) Q∗ = Q − f0g, R∗ = R − f0g, C∗ = C − f0g with multiplication 3) GLn(R) = fA 2 Mn(R) j det(A) 6= 0g with matrix multiplication (the n × n general linear group) 4) SLn(R) = fA 2 Mn(R) j det(A) = 1g with matrix multiplication (the n × n special linear group) 5) Let U = be any set and let Perm(U) := ff : U ! U j f is a bijectiong Perm(U) with composition of functions is a group (the group of permu- tations of U) Note. If U = f1; 2; : : : ; ng then Perm(U) is called the symmetric group on n letters and it is denoted by Sn. 2 7) Let T = an equilateral triangle GT = fI;R1;R2;S1;S2;S3g I R2 R1 S S 1 S2 3 GT = the group of symmetries of T . 1.9 Proposition (Cancellation Law). If G is a group, x; y; x 2 G and xy = xz then y = z. Proof. xy = xz x−1xy = x−1xz y = z 1.10 Note. The cancellation law does not hold for monoids. E.g. in M2(R) take 1 0 0 0 0 0 A = ;B = ;C = 0 0 0 1 0 0 Then AB = AC but A 6= C. 3 2 Subgroups 2.1 Definition. If G is a group then a subgroup of G is a subset H ⊆ G such that (i) e 2 H; (ii) if x; y 2 H then xy 2 H; (iii) if x 2 H then x−1 2 H. 2.2 Note. A subgroup of a group is by itself a group. 2.3 Examples. 1) If G is a group then G, feg are subgroups of G 2) Z is a subgroup of Q, which is a subgroup of R, which is a subgroup of C. 3) SLn(R) is a subgroup of GLn(R) 4) H = fI;R1;R2g is a subgroup of GT T 2.4 Note. If fHigi2I is a family of subgroups of G then i2I Hi is also a subgroup of G. 2.5 Definition. If G is a group and S is a subset of G then denote hSi = the smallest subgroup of G that contains S hSi is the subgroup of G generated by the set S. 2.6 Proposition. If S ⊆ G then hSi consists of all elements of the form ±1 ±1 ±1 x1 x2 ····· xk where x1; : : : ; xk 2 S. 4 Proof. Exercise. 2.7 Definition. A set S ⊆ G generates G if hSi = G. 2.8 Example. S = fS1;S2g generates GT . 2.9 Definition. A group G is finitely generated if it is generated by some finite subset S ⊆ G. 2.10 Note. • Every finite group is finitely generated. • Some infinite groups are finitely generated; e.g. Z = h1i. 2.11 Definition. A group G is cyclic if G = hai for some a 2 G 2.12 Note. If G is cyclic, G = hai then every element g 2 G is of the form g = an for some n 2 Z (where a−n := (a−1)n, a0 = e). 2.13 Examples. 1) Z = h1i is cyclic. 2) H := fI;R1;R2g ⊆ GT is cyclic: H = hR1i and H = hR2i 5 3 Homomorphisms of groups 3.1 Definition. Let G, H be groups. A function f : G ! H is a group homo- morphism if for any a; b 2 G we have f(ab) = f(a)f(b) 3.2 Proposition. If f : G ! H is a homomorphism of groups and eG, eH denote identity elements in, respectively, G and H then (i) f(eG) = eH (ii) f(a−1) = f(a)−1 for any a 2 G. Proof. (i) We have f(eG) = f(eG · eG) = f(eG) · f(eG) −1 Multiplying this equation by f(eG) we obtain eH = f(eG). (ii) Since by (i) we have f(eG) = eH therefore −1 −1 f(a) · f(a ) = f(a · a ) = f(eG) = eH It is now enough to multiply this equation from the left by f(a)−1. 3.3 Definition. A homomorphism f : G ! H is an isomorphism if there is a homomorphism g : H ! G such that g ◦ f = idG and f ◦ g = idH . 3.4 Proposition. A map f : G ! H is an isomorphism of groups iff f is a homomorphism and a bijection. Proof. Exercise. 3.5 Definition. If there exists an isomorphism f : G ! H then we say that the groups G and H are isomorphic and we write G ∼= H. 6 3.6 Definition. A homomorphism f : G ! G is called an endomorphism of G. An isomorphism f : G ! G is called an automorphism of G. 3.7 Examples. 1) idG : G ! G is an automorphism of G. 2) f : G ! G, f(g) = e 8g2G is an endomorphism of G. 3) If f : G ! H, g : H ! K are homomorphisms then so is g ◦ f : G ! K. 4) For g 2 G define −1 cg : G ! G; cg(a) := gag Check: cg is an automorphism of G. Automorphisms of this form are called inner automorphisms of G. Note. If G is an abelian group then cg = idG for all g 2 G. ∗ 5) Recall: GLn(R) = fA 2 Mn j det(A) 6= 0g, R = R − f0g We have the determinant function: ∗ det: GLn(R) ! R Since det(AB) = det(A) · det(B) this function is a homomorphism. 6) Let G ⊆ GL2(R) 1 r G := r 2 R 0 1 G is a subgroup of GL2(R): 1 r 1 s 1 r + s · = 0 1 0 1 0 1 1 r−1 1 −r = 0 1 0 1 We have homomorphisms: f : R ! G and g : R ! G 7 where 1 r 1 r f(r) = ; g = r 0 1 0 1 ∼ Since g ◦ f = idG, f ◦ g = R we get G = R. 3.8 Definition. If G is a group then jGj := the number of elements of G jGj is called the order of G. 3.9 Example. jGT j = 6, jZj = 1. 3.10 Note. If G ∼= H then jGj = jHj. 8 4 The kernel and the image of a homomorphism 4.1 Proposition. Let f : G ! H be a homomorphism. 1) If G0 is a subgroup of G then f(G0) is a subgroup of H. 2) If H0 is a subgroup of H then f −1(H0) is a subgroup of G. Proof. Exercise. 4.2 Definition. If f : G ! H is a homomorphism then • the image of f is the subgroup Im(f) := f(G) ⊆ H • the kernel of f is the subgroup −1 Ker(f) := f (eH ) ⊆ G 4.3 Note. f : G ! H is an epimorphism (onto) iff Im(f) = H. 4.4 Proposition. f : G ! H is a monomorphism (1-1) iff Ker(f) = feGg Proof. ()) We have f(eG) = eH . Thus if f is 1-1 then f(g) = eH only if g = eH . In other words we have then Ker(f) = feH g. (() Assume that Ker(f) = feGg and let f(a) = f(b) for some a; b 2 G. We have: −1 −1 f(ab ) = f(a)f(b) = eH −1 −1 so ab 2 Ker(f). Therefore ab = eG, and so a = b. 9 4.5 Problem. Let G be a group, and let H be a subgroup of G. Is there a homomorphism f : G ! K such that Ker(f) = H? 4.6 Note. The dual problem is trivial: if H is a subgroup of G then we have the inclusion homomorphism i: H,! G and Im(i) = H.
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