Theorems and Definitions in Group Theory Shunan Zhao Contents 1 Basics of a group 3 1.1 Basic Properties of Groups . 3 1.2 Properties of Inverses . 3 1.3 Direct Product of Groups . 4 2 Equivalence Relations and Disjoint Partitions 4 3 Elementary Number Theory 5 3.1 GCD and LCM . 5 3.2 Primes and Euclid's Lemma . 5 4 Exponents and Order 6 4.1 Exponents . 6 4.2 Order . 6 5 Integers Modulo n 7 5.1 Integers Modulo n .............................. 7 5.2 Addition and Multiplication of Zn ...................... 7 6 Subgroups 7 6.1 Properties of Subgroups . 7 6.2 Subgroups of Z ................................ 8 6.3 Special Subgroups of a Group G ....................... 8 6.4 Creating New Subgroups from Given Ones . 8 7 Cyclic Groups and Their Subgroups 9 7.1 Cyclic Subgroups of a Group . 9 7.2 Cyclic Groups . 9 7.3 Subgroups of Cyclic Groups . 9 7.4 Direct Product of Cyclic Groups . 10 8 Subgroups Generated by a Subset of a Group G 10 9 Symmetry Groups and Permutation Groups 10 9.1 Bijections . 10 9.2 Permutation Groups . 11 9.3 Cycles . 11 9.4 Orbits of a Permutation . 12 9.5 Generators of Sn ............................... 12 1 10 Homorphisms 13 10.1 Homomorphisms, Epimorphisms, and Monomorphisms . 13 10.2 Classification Theorems . 14 11 Cosets and Lagrange's Theorem 15 11.1 Cosets . 15 11.2 Lagrange's Theorem . 15 12 Normal Subgroups 16 12.1 Introduction to Normal Subgroups . 16 12.2 Quotient Groups . 16 13 Product Isomorphism Theorem and Isomorphism Theorems 17 13.1 Product Isomorphism Theorem . 17 13.2 Isomorphism Theorems . 18 2 1 Basics of a group 1.1 Basic Properties of Groups Definition 1.1.1 (Definition of a Group). A set G is a group if and only if G satisfies the following: 1. G has a binary relation · : G × G ! G so that 8g; h 2 G; g · h 2 G. We write ·(g; h) = g · h. (Closure.) 2. 8g; h; k 2 G; g · (h · k) = (g · h) · k. (Associative.) 3. 9e 2 G; s.t. e · a = a = a · e. e is called an identity element of G. 4. 8g 2 G; 9g−1 2 G; s.t. g · g−1 = e = g−1 · g. g−1 is called an inverse of g. Definition 1.1.2. If 8g; h 2 G, we also have g · h = h · g, then we say that G is an abelian group. Theorem 1.1.1. Let G be a group. Then, • 8a 2 G; aG = G = Ga, where Ga = fga : g 2 Gg and aG = fag : g 2 Gg • If a; x; y 2 G, then ax = ay =) x = y. • If a; x; y 2 G, then xa = ya =) x = y. Theorem 1.1.2. G is a group. Then: • G has only one identity element. • Each g 2 G has only one inverse g−1. 1.2 Properties of Inverses Theorem 1.2.1. G is a group. • If g 2 G, then (g−1)−1 = g. • If g; h 2 G, then (gh)−1 = h−1g−1. Theorem 1.2.2. Let G be a set with the following axioms: 1. Closure: g; h 2 G =) gh 2 G. 2. Associativity: 8g; h; k 2 G; g(hk) = (gh)k. 3. 9e 2 G; 8g 2 G; eg = g.(e is a left identity.) 4. 8g 2 G; 9 ∗ g 2 G; ∗gg = e.(∗g is a left inverse.) Then, G is a group. The same applies for a right inverse and a right identity. 3 1.3 Direct Product of Groups Theorem 1.3.1. Let G and H be two groups. Define the direct product of G and H as G × H = f(g; h): g 2 G; h 2 Hg. Then, G × H is a group with the component-wise binary operation. Definition 1.3.1. Let C = R × R = f(a; b): a; b 2 Rg. We define addition and multiplication in C as: • Addition: We want to use the component wise addition from (R; +). Thus, (a; b)+ (c; d) = (a + b; c + d). Then, 0 = (0; 0) and (a; b)−1 = −(a; b) = (−a; −b). • Multiplication: Define multiplication by (a; b)(c; d) = (ac − bd; ad + bc). Then, the multiplicative identity is (1; 0). Furthermore, define i = (0; 1). Then, the −1 z¯ multiplicative inverse is z = jzj2 . Definition 1.3.2. Define H = C × C = f(z; w): z; w 2 Cg. We define addtion component-wise and multiplication by (z; w)(u; v) = (zu − wv;¯ zv − wu¯). The multi- −1 h¯ plicative identity is (1; 0) and h = jhj2 . 2 Equivalence Relations and Disjoint Partitions Definition 2.0.3. Let X be a set and R be a relation on X. R is called an equivalence relation on X if and only if the following axioms hold: 1. For each x 2 X; xRx (R is reflexive). 2. 8x; y 2 X, if xRy, then yRx (R is symmetric). 3. If xRy and yRz, then xRz (R is transitive). Definition 2.0.4. 8x 2 X; R[x] = fy 2 X : yRxg is called an equivalence class. Theorem 2.0.2. Let R be an equivalence relation on X. Then, R[x] = R[z] () xRz Theorem 2.0.3. Let R be an equivalence relation on a set X. Let DR = fR[x]: x 2 Xg. Then, DR has the following properties: 1. R[x] 6= ;, since xRx =) x 2 R[x]. 2. If R[x] \ R[y] 6= ;, then R[x] = R[y]. [ 3. X = R[x]. x2X Definition 2.0.5. Let X be a set. The power set of X is P(X) = fS : S ⊆ Xg: Definition 2.0.6 (Disjoint Partition). A subset D ⊆ P(X) is a disjoint partition of X if and only if D satisfies the following axioms: 1. 8D 2 D;D 6= ;. 2. If D; D~ 2 D and D \ D~ 6= ;, then D = D~. 4 [ 3. X = D. D2D Corollary 2.0.4. If R is an equivalence relation on a set X, then DR = fR[x]: x 2 Xg is a disjoint partition of X. Theorem 2.0.5. Let D be a disjoint partition of a set X. Define a relation on X,RD as follows: xRDy () 9D 2 D so that x; y 2 D: Then, RD is an equivalence relation. 3 Elementary Number Theory 3.1 GCD and LCM Axiom 3.1.1 (The Well Ordering Principle). Every non-empty subset of N has a smallest element. Theorem 3.1.1 (Division Algorithm). Let a; b 2 Z; b 6= 0. Then, 9!q; r 2 Z s.t. a = bq + r; 0 ≤ r < jbj. Definition 3.1.1. Let a; b 2 Z, not both zero. Then, c > 0; c 2 N is a greatest common divisor of a and b () c satisfies the following properties (a; b 2 Z; a 6= 0 6= b): 1. cja and cjb. 2. If xja and xjb, then xjc. Theorem 3.1.2. There exists a GCD of a and b, say c, and c = λa + µb, λ, µ 2 Z. Furthermore, c is smallest n 2 N s.t. n = λa + µb. Definition 3.1.2. Let a; b 2 Z; a > 0; b > 0. d 2 Z and d > 0 is a least common multiple of a and b if and only if d satisfies the folloing properties: 1. ajd and bjd. 2. If ajx and bjx, then djx. ab Theorem 3.1.3. Assume a; b 2 ; a > 0; b > 0. Then, d = is the LCM of a Z gcd(a; b) and b. 3.2 Primes and Euclid's Lemma Definition 3.2.1. p 2 N; p is a prime () njp =) n = 1 or n = p. Theorem 3.2.1. Let p 2 N. p is a prime if and only if 8n 2 N; pjn or gcd(n; p) = 1 (relatively prime). Theorem 3.2.2. For a; b; c 2 N, if ajbc and gcd(a; b) = 1, then ajc. Corollary 3.2.3 (Euclid's Lemma). If p is a prime and pjab, then pja or pjb. 5 4 Exponents and Order 4.1 Exponents Definition 4.1.1. G is a group and a 2 G. We define a0 = e, a1 = a and an+1 = an · a, for n 2 N+. Also, if m > 0, define a−m = (a−1)m. Theorem 4.1.1 (Properties of Exponents). G is a group and a; b 2 G and m; n 2 N. Then, we have 1. en = e 2. am+n = aman 3. (am)n = amn 4. Let ab = ba. Then, abn = bna and (ab)n = anbn = bnan. 5. a−m = (am)−1 6. If 0 ≤ m ≤ n, then an−m = ana−m. 7. (a−1ba)n = a−1bna 4.2 Order Definition 4.2.1. G is a group and a 2 G. If there is a positive integer n > 0 s.t. an = e, we say a has finite order. If a has finite order, then the smallest n > 0 s.t. an = e is called the order of a. We write o(a) as the order of a. If 8n 2 N+; an 6= e, then we say a has infinite order and we write o(a) = 1. If o(a) 6= 1, then a has finite order. Theorem 4.2.1. Let G be a group and a 2 G; m > 0.