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.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.4: Cyclic Groups
Problem 5E: The elements of the multiplicative group G of 33 permutation matrices are given in Exercise 35 of...
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1a6a995-c8d0-4d4f-9e6b-a9cb754faf07%2F67d951de-fd70-46e4-8472-db0f73d96cc0%2Fvcuojin_processed.jpeg&w=3840&q=75)
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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