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Activity 25.22. One advantage of decomposing a permutation as a product of disjoint cycles is that (a1 a2 ak) and 3 = (b₁ b2 ... bm) disjoint cycles commute. To verify this property, let a = be disjoint cycles in Sn. (a) What can we say about the sets A = {a₁, a2,..., ak} and B = {a₁, a2, ..., ak} and B = {b₁,b2, ..., bm}? (b) What is a (b;) for any 1 ≤ j≤m? What is 3(a) for any 1 ≤ i ≤k? Explain. (c) If r = {1,2,...,n}\(AUB), what are ax(r) and 3(r)? Explain. (d) Show that (a) (t) = (Ba)(t) for every t = {1, 2,..., n} by considering cases.

Activity 25.22. One advantage of decomposing a permutation as a product of disjoint cycles is that (a1 a2 ak) and 3 = (b₁ b2 ... bm) disjoint cycles commute. To verify this property, let a = be disjoint cycles in Sn. (a) What can we say about the sets A = {a₁, a2,..., ak} and B = {a₁, a2, ..., ak} and B = {b₁,b2, ..., bm}? (b) What is a (b;) for any 1 ≤ j≤m? What is 3(a) for any 1 ≤ i ≤k? Explain. (c) If r = {1,2,...,n}\(AUB), what are ax(r) and 3(r)? Explain. (d) Show that (a) (t) = (Ba)(t) for every t = {1, 2,..., n} by considering cases.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.5: The Binomial Theorem
Problem 15E
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Abstract Algebra (Hodge, Jonathan K) Activity 25.22. One advantage of decomposing a permutation as a product of ,dots, a 1) and B = ([b₁,b₂, dots, disjoint cycles is that disjoint cycles commute. To verify this property, let ax = ([a, , a₂₁ adots,a} and b]) be disjoint cycles in S. (a) What can we say about the sets A = {a_ m 11 ...b} ? (b) What is a(b) for any 1 ≤j s m? What is B(a) for any 1 ≤ i ≤k ? Explain. (c) If rin (AUB), what are o(r) and B(r) ? Explain. (d) Show that (xß)(t) = (Ba)(t) for every tin{1,2, dots,n} by B = {b b₁,b₂₁ 1,2,...,n 1 considering cases. V Activity 25.22. One advantage of decomposing a permutation as a product of disjoint cycles is that disjoint cycles commute. To verify this property, let a = (a₁ a₂ .... ak) and 3 = (b1 b2 bm) be disjoint cycles in Sn. (a) What can we say about the sets A = = {a1, a2,..., ak} and B = {bi, b2, ..., bm}? (b) What is a (b) for any 1 ≤ j≤m? What is B(a) for any 1 ≤ i ≤k? Explain. (c) If r € {1,2,...,n}\(AUB), what are a(r) and B(r)? Explain. (d) Show that (aß)(t) = (Ba) (t) for every te {1, 2,...., n} by considering cases.
Transcribed Image Text:Abstract Algebra (Hodge, Jonathan K) Activity 25.22. One advantage of decomposing a permutation as a product of ,dots, a 1) and B = ([b₁,b₂, dots, disjoint cycles is that disjoint cycles commute. To verify this property, let ax = ([a, , a₂₁ adots,a} and b]) be disjoint cycles in S. (a) What can we say about the sets A = {a_ m 11 ...b} ? (b) What is a(b) for any 1 ≤j s m? What is B(a) for any 1 ≤ i ≤k ? Explain. (c) If rin (AUB), what are o(r) and B(r) ? Explain. (d) Show that (xß)(t) = (Ba)(t) for every tin{1,2, dots,n} by B = {b b₁,b₂₁ 1,2,...,n 1 considering cases. V Activity 25.22. One advantage of decomposing a permutation as a product of disjoint cycles is that disjoint cycles commute. To verify this property, let a = (a₁ a₂ .... ak) and 3 = (b1 b2 bm) be disjoint cycles in Sn. (a) What can we say about the sets A = = {a1, a2,..., ak} and B = {bi, b2, ..., bm}? (b) What is a (b) for any 1 ≤ j≤m? What is B(a) for any 1 ≤ i ≤k? Explain. (c) If r € {1,2,...,n}\(AUB), what are a(r) and B(r)? Explain. (d) Show that (aß)(t) = (Ba) (t) for every te {1, 2,...., n} by considering cases.
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