Milne - Group Theory.. - Free

More documents

Recommendations

Info

44 4 GROUPS ACTING ON SETS Since the orbits of an element α of S n form a partition of {1,...,n}, we can attach to eachsuchα a partition of n. For example, if α =(i 1 ...i n1 ) ···(l 1 ...l nk ), (disjoint cycles) 1
Permutation groups 45 Note that A 4 contains exactly 3 elements of order 2, namely those of type 2 + 2, and that together with 1 they form a subgroup V . This group is a union of conjugacy classes, and is therefore a normal subgroup of S 4 . Theorem 4.29 (Galois). The group A n is simple if n ≥ 5 Remark 4.30. For n =2,A n is trivial, and for n =3,A n is cyclic of order 3, and hence simple; for n = 4 it is nonabelian and nonsimple (it contains the normal, even characteristic, subgroup V — see above). Lemma 4.31. Let N be a normal subgroup of A n (n ≥ 5); ifN contains a cycle of length three, then it contains all cycles of length three, and so equals A n (by 4.24). Proof. Let γ be the cycle of length three in N, andletα be a second cycle of length three in A n . We know from (4.26) that α = gγg −1 for some g ∈ S n .Ifg ∈ A n ,then this shows that α is also in N. If not, because n ≥ 5, there exists a transposition t ∈ S n disjoint from α. Thentg ∈ A n and and so again α ∈ N. α = tαt −1 = tgγg −1 t −1 , The next lemma completes the proof of the Theorem. Lemma 4.32. Every normal subgroup N of A n , n ≥ 5, N ≠1, contains a cycle of length 3. Proof. Let α ∈ N, α ≠1. Ifα is not a 3-cycle, we shall construct another element α ′ ∈ N, α ′ ≠ 1, which fixes more elements of {1, 2,...,n} than does α. Ifα ′ is not a 3-cycle, then we can apply the same construction. After a finite number of steps, we arrive at a 3-cycle. Suppose α is not a 3-cycle. When we express it as a product of disjoint cycles, either it contains a cycle of length ≥ 3 or else it is a product of transpositions, say (i) α =(i 1 i 2 i 3 ...) ··· or (ii) α =(i 1 i 2 )(i 3 i 4 ) ···. In the first case, α moves two numbers, say i 4 , i 5 , other than i 1 , i 2 , i 3 , because α ≠(i 1 i 2 i 3 ), (i 1 ...i 4 ). Let γ =(i 3 i 4 i 5 ). Then α 1 = df γαγ −1 =(i 1 i 2 i 4 ...) ··· ∈ N, and is distinct from α (because it acts differently on i 2 ). Thus α ′ = df α 1 α −1 ≠ 1, but α ′ = γαγ −1 α −1 fixes i 2 and all elements other than i 1 , ..., i 5 fixed by α — it therefore fixes more elements than α. In the second case, form γ, α 1 , α ′ as in the first case with i 4 as in (ii) and i 5 any element distinct from i 1 ,i 2 ,i 3 ,i 4 .Thenα 1 =(i 1 i 2 )(i 4 i 5 ) ··· is distinct from α because it acts differently on i 4 .Thusα ′ = α 1 α −1 ≠ 1, but α ′ fixes i 1 and i 2 ,andallelements ≠ i 1 , ..., i 5 not fixed by α — it therefore fixes at least one more element than α.
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 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: Permutation groups 43 Note that the
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

Milne - Group Theory.. - Free

Create successful ePaper yourself

Delete template?

Save as template?