Milne - Group Theory.. - Free

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6 1 BASIC DEFINITIONS 12: C 12 , C 2 × C 6 , C 2 × S 3 , A 4 , C 3 ⋊ C 4 (see 3.13 below). 13: C 13 . 14: C 14 , D 7 . 15: C 15 . 16: (14 groups) General rules: For eachprime p, there is only one group (up to isomorphism), namely C p (see 1.17 below), and only two groups of order p 2 ,namely,C p × C p and C p 2 (see 4.17). For the classification of the groups of order 6, see 4.21; for order 8, see 5.15; for order 12, see 5.14; for orders 10, 14, and 15, see 5.12. Roughly speaking, the more high powers of primes divide n, themoregroupsof order n youexpect. Infact,iff(n) is the number of isomorphism classes of groups of order n, then f(n) ≤ n ( 2 27 +o(1))e(n)2 where e(n) is the largest exponent of a prime dividing n and o(1) → 0ase(n) →∞ (see Pyber, Ann. of Math., 137 (1993) 203–220). By 2001, a complete irredundant list of groups of order ≤ 2000 had been found — up to isomorphism, there are 49,910,529,484 (Besche, Hans Ulrich; Eick, Bettina; O’Brien, E. A. The groups of order at most 2000. Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1–4 (electronic)). Multiplication tables A law of composition on a finite set can be described by its multiplication table: 1 a b c ... 1 1 a b c ... a a a 2 ab ac . . . b b ba b 2 bc ... c c ca cb c 2 ... . . . . . Note that, if the law of composition defines a group, then, because of the cancellation laws, eachrow (and eachcolumn) is a permutation of the elements of the group. This suggests an algorithm for finding all groups of a given finite order n, namely, list all possible multiplication tables and check the axioms. Except for very small n, this is not practical! There are n 3 possible multiplication tables for a set with n elements, and so this quickly becomes unmanageable. Also checking the associativity law from a multiplication table is very time consuming. Note how few groups there are. The 12 3 = 1728 possible multiplication tables for a set with12 elements give only 5 isomorphism classes of groups.
Homomorphisms 7 Homomorphisms Definition 1.9. A homomorphism from a group G to a second G ′ is a map α: G → G ′ suchthat α(ab) =α(a)α(b) for all a, b ∈ G. Note that an isomorphism is simply a bijective homomorphism. Remark 1.10. Let α be a homomorphism. Then α(a m )=α(a m−1 · a) =α(a m−1 ) · α(a), and so, by induction, α(a m )=α(a) m , m ≥ 1. Moreover α(1) = α(1 × 1) = α(1)α(1), and so α(1) = 1 (apply 1.2a). Also aa −1 =1=a −1 a ⇒ α(a)α(a −1 )=1=α(a −1 )α(a), and so α(a −1 )=α(a) −1 . From this it follows that α(a m )=α(a) m all m ∈ Z. We saw above that each row of the multiplication table of a group is a permutation of the elements of the group. As Cayley pointed out, this allows one to realize the group as a group of permutations. Theorem 1.11 (Cayley theorem). There is a canonical injective homomorphism α: G → Sym(G). Proof. For a ∈ G, define a L : G → G to be the map x ↦→ ax (left multiplication by a). For x ∈ G, (a L ◦ b L )(x) =a L (b L (x)) = a L (bx) =abx =(ab) L (x), and so (ab) L = a L ◦ b L .As1 L = id, this implies that a L ◦ (a −1 ) L =id=(a −1 ) L ◦ a L , and so a L is a bijection, i.e., a L ∈ Sym(G). Hence a ↦→ a L is a homomorphism G → Sym(G), and it is injective because of the cancellation law. Corollary 1.12. Afinite group of order n can be identified with a subgroup of S n . Proof. Numbertheelementsofthegroupa 1 ,...,a n . Unfortunately, when G has large order n, S n is too large to be manageable. We shall see later (4.20) that G can often be embedded in a permutation group of much smaller order than n!.
Page 1 and 2: GROUP THEORY J.S. Milne Abstract Th
Page 3 and 4: ii Notations. We use the standard (
Page 5 and 6: 1 1 Basic Definitions Definitions D
Page 7 and 8: Subgroups 3 group. For example GL n
Page 9: Groups of small order 5 and G can b
Page 13 and 14: Normal subgroups 9 Example 1.17. If
Page 15 and 16: Quotients 11 Quotients The kernel o
Page 17 and 18: 13 2 Free Groups and Presentations
Page 19 and 20: Free groups 15 We say two words w,
Page 21 and 22: Generators and relations 17 Lemma 2
Page 23 and 24: Finitely presented groups 19 group
Page 25 and 26: 21 3 Isomorphism Theorems. Extensio
Page 27 and 28: Direct products 23 (b) H 1 ∩ H 2
Page 29 and 30: Automorphisms of groups 25 and so w
Page 31 and 32: Semidirect products 27 Remark 3.12.
Page 33 and 34: Semidirect products 29 Write G = N
Page 35 and 36: Extensions of groups 31 Extensions
Page 37 and 38: Exercises 13-19 33 Part A has been
Page 39 and 40: General definitions and results 35
Page 45 and 46: Permutation groups 41 Permutation g
Page 47 and 48: Permutation groups 43 Note that the
Page 49 and 50: Permutation groups 45 Note that A 4
Page 51 and 52: The Todd-Coxeter algorithm. 47 The
Page 53 and 54: Exercises 20-33 49 Proof. =⇒: The
Page 55 and 56: 51 5 The Sylow Theorems; Applicatio
Page 57 and 58: The Sylow theorems 53 the highest p
Page 59 and 60: Applications 55 (a) α(V i ) ⊂ V
Page 61 and 62:
Applications 57 Now assume that P i
Page 63 and 64:
59 6 Normal Series; Solvable and Ni
Page 65 and 66:
Solvable groups 61 Similarly G/H 1
Page 67 and 68:
Solvable groups 63 (b) Let N be a n
Page 69 and 70:
Nilpotent groups 65 Nilpotent group
Page 71 and 72:
Nilpotent groups 67 Corollary 6.16.
Page 73 and 74:
Groups with operators 69 Example 6.
Page 75 and 76:
Further reading 71 Theorem 6.29 (Kr
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73 10. The hint gives ab 3 a −1 =
Page 79 and 80:
75 Deduce that #C G (α) = 3, and s
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77 B Review Problems 34. Prove that
Page 83 and 84:
79 (e) Describe all subgroups of G
Page 85 and 86:
81 73. (a) Prove that subgroups A a
Page 87 and 88:
Solutions 83 Solutions 1. (a) False
Page 89 and 90:
Index action doubly transitive, 36
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