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26 3 ISOMORPHISM THEOREMS. EXTENSIONS. Remark 3.10. It will be useful to have a description of (Z/nZ) × =Aut(C n ). If n = p r 1 1 ···prs s is the factorization of n into powers of distinct primes, then the Chinese Remainder Theorem (Dummit and Foote 1991, 7.6, Theorem 17) gives us an isomorphism Z/nZ ∼ = Z/p r 1 1 Z ×···×Z/p rs s Z, m mod n ↦→ (m mod p r 1 1 ,...,m mod p rs s ), which induces an isomorphism (Z/nZ) × ≈ (Z/p r 1 1 Z) × ×···×(Z/p rs s Z)× . Hence we need only consider the case n = p r , p prime. Suppose first that p is odd. The set {0, 1,...,p r − 1} is a complete set of representatives for Z/p r Z,and 1 of these elements are divisible by p. Hence p (Z/pr Z) × has order p r − pr = p pr−1 (p − 1). Because p − 1andp r are relatively prime, we know from (1.3d) that (Z/p r Z) × is isomorphic to the direct product of a group A of order p − 1 and a group B of order p r−1 .Themap (Z/p r Z) × ↠ (Z/pZ) × = F × p , induces an isomorphism A → F × p ,andF× p , being a finite subgroup of the multiplicative group of a field, is cyclic (FT, Exercise 3). Thus (Z/p r Z) × ⊃ A = 〈ζ〉 for some element ζ of order p − 1. Using the binomial theorem, one finds that 1 + p has order p r−1 in (Z/p r Z) × , and therefore generates B. Th us (Z/p r Z) × is cyclic, withgenerator ζ · (1 + p), and every element can be written uniquely in the form On the other hand, ζ i · (1 + p) j , 0 ≤ i
Semidirect products 27 Remark 3.12. (a) Consider a group G and a normal subgroup H. An inner automorphism of G restricts to an automorphism of H, which may be outer (for an example, see 3.16f). Thus a normal subgroup of H need not be a normal subgroup of G. However, a characteristic subgroup of H will be a normal subgroup of G. Also a characteristic subgroup of a characteristic subgroup is a characteristic subgroup. (b) The centre Z(G) ofG is a characteristic subgroup, because zg = gz all g ∈ G ⇒ α(z)α(g) =α(g)α(z) allg ∈ G, and as g runs over G, α(g) also runs over G. Expect subgroups witha general group-theoretic definition to be characteristic. (c) If H is the only subgroup of G of order m, then it must be characteristic, because α(H) is again a subgroup of G of order m. (d) Every subgroup of a commutative group is normal but not necessarily characteristic. For example, a subspace of dimension 1 in G = F 2 p will not be stable under GL 2 (F p ) and hence is not a characteristic subgroup. Semidirect products Let N be a normal subgroup of G. Eachelement g of G defines an automorphism of N, n ↦→ gng −1 , and so we have a homomorphism θ : G → Aut(N). If there exists a subgroup Q of G suchthat G → G/N maps Q isomorphically onto G/N, then I claim that we can reconstruct G from the triple (N,Q,θ|Q). Indeed, any g ∈ G can be written in a unique fashion g = nq, n ∈ N, q ∈ Q — q is the unique element of Q representing g in G/N, andn = gq −1 .Thus,wehave a one-to-one correspondence (of sets) If g = nq and g ′ = n ′ q ′ ,then G 1−1 ↔ N × Q. gg ′ = nqn ′ q ′ = n(qn ′ q −1 )qq ′ = n · θ(q)(n ′ ) · qq ′ . Definition 3.13. AgroupG is said to be a semidirect product of the subgroups N and Q, written N ⋊ Q, ifN is normal and G → G/N induces an isomorphism Q ≈ → G/N. Equivalent condition: N and Q are subgroups of G suchthat (i) N ⊳ G; (ii) NQ = G; (iii) N ∩ Q = {1}. Note that Q need not be a normal subgroup of G.
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 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: Automorphisms of groups 25 and so w
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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