Milne - Group Theory.. - Free

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16 2 FREE GROUPS AND PRESENTATIONS Corollary 2.5. Every group is a quotient of a free group. Proof. Choose a set X of generators for G (e.g., X = G), and let F be the free group generated by X. According to (2.3), the inclusion X ↩→ G extends to a homomorphism F → G, and the image, being a subgroup containing X, mustequal G. The free group on the set X = {a} is simply the infinite cyclic group C ∞ generated by a, but the free group on a set consisting of two elements is already very complicated. I now discuss, without proof, some important results on free groups. Theorem 2.6 (Nielsen-Schreier). 8 Subgroups of free groups are free. The best proof uses topology, and in particular covering spaces—see Serre, Trees, Springer, 1980, or Rotman 1995, Theorem 11.44. Two free groups FX and FY are isomorphic if and only if X and Y have the same number of elements 9 . Thus we can define the rank of a free group G to be the number of elements in (i.e., cardinality of) a free generating set, i.e., subset X ⊂ G such that the homomorphism FX → G given by (2.3) is an isomorphism. Let H be a finitely generated subgroup of a free group F . Then there is an algorithm for constructing from any finite set of generators for H a free finite set of generators. If F has rank n and (F : H) =i
Generators and relations 17 Lemma 2.7. The normal subgroup generated by S ⊂ G is 〈 ⋃ g∈G gSg−1 〉. Consider a set X and a set R of words made up of symbols in X ′ . Eachelement of R represents an element of the free group FX, and the quotient G of FX by the normal subgroup generated by these elements is said to have X as generators and R as relations. One also says that (X, R) isapresentation for G, G = 〈X|R〉, andthat R is a set of defining relations for G. Example 2.8. (a) The dihedral group D n has generators σ, τ and defining relations σ n ,τ 2 ,τστσ. (See 2.10 below for a proof.) (b) The generalized quaternion group Q n , n ≥ 3, has generators a, b and relations 11 a 2n−1 =1,a 2n−2 = b 2 , bab −1 = a −1 .Forn = 3 this is the group Q of (1.8c). In general, it has order 2 n (for more on it, see Exercise 8). (c) Two elements a and b in a group commute if and only if their commutator [a, b] = df aba −1 b −1 is 1. The free abelian group on generators a 1 ,...,a n has generators a 1 ,a 2 ,...,a n and relations [a i ,a j ], i ≠ j. For the remaining examples, see Massey, W., Algebraic Topology: An Introduction, Harbrace, 1967, which contains a good account of the interplay between group theory and topology. For example, for many types of topological spaces, there is an algorithm for obtaining a presentation for the fundamental group. (d) The fundamental group of the open disk with one point removed is the free group on σ where σ is any loop around the point (ibid. II 5.1). (e) The fundamental group of the sphere with r points removed has generators σ 1 , ..., σ r (σ i is a loop around the i th point) and a single relation σ 1 ···σ r =1. (f) The fundamental group of a compact Riemann surface of genus g has 2g generators u 1 ,v 1 , ..., u g ,v g and a single relation (ibid. IV Exercise 5.7). u 1 v 1 u −1 1 v−1 1 ···u g v g u −1 g v−1 g =1 Proposition 2.9. Let G be the group defined by the presentation (X, R). For any group H and map (of sets) X → H sending each element of R to 1 (inanobvious sense), there exists a unique homomorphism G → H making the following diagram commute: X >G ❅ ❅❅❘ ∨ H. 11 Strictly speaking, I should say the relations a 2n−1 , a 2n−2 b −2 , bab −1 a.
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 and 10: Groups of small order 5 and G can b
Page 11 and 12: Homomorphisms 7 Homomorphisms Defin
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: Free groups 15 We say two words w,
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
Page 77 and 78:
73 10. The hint gives ab 3 a −1 =
Page 79 and 80:
75 Deduce that #C G (α) = 3, and s
Page 81 and 82:
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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