6. Show that a cycle of odd length is an even permutation, and a cycle of even length is an oddpermutation. Hence determine which of the following members of S10 are even, or odd:(i) (19 3) (2 6) (4 5 10 ) (78)1 9 3 2 6) (4 5 10 7 8),(iii) 1 9 3 2 6 4 5 10).

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Answered: . Show that a cycle of odd length is an…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.1: Finite Permutation Groups

Problem 1E: Exercises 1. Express each permutation as a product of disjoint cycles and find the orbits of each...

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Exercises 1. Express each permutation as a product of disjoint cycles and find the orbits of each permutation. a. b. c. d. e. f. g. h.
14. Consider the permutations a = (5 21 4 3), B = (1 3)(2 4 6), y = (12 4)(3 5 6) of {1,2, 3, 4, 5, 6} expressed in cycle notation. Which one of the following is correct? (a) a and y are odd, and B is even. (b) a and y are even, and ß is odd. (c) a and B are even, and y is odd. (d) a and B are odd, and y is even. (e) B and y are even, and a is odd.
The permutations p = (3 5 4), q = (1 5 2 3) and r = (1 5 2) are elements of S5. (a) Find each of the following as a permutation in cycle form: wold wode ei doang poq, poqop-¹, q². (b) State the order and parity of each of po q and q². (c) Determine, in cycle form, two permutations in S5 that conjugate p to r.
If o is a k-cycle then the order of o is k. On the otherhand, if 7 is a product of disjoint cycles o1, 02, 03 .m such that the order of o; is k; then the order of 7 is given by lem(k1, k2, kg, ..., km). Find the order each of the following. (a) (1 32 4) e S4. (b) (1 4 3)(2 5 6 7) E S7. (c) (1 43 6)(2 9)(5 7 8) E Sg. (d) (1 7)(2 6)(3 5 4) o (1 2 3)(4 5 6 7) e S,
Suppose α is a single cycle of length m and β is a single cycle of length n. What can be said about the order of the element αβ if α and β are disjoint cycles? If the cycles α and β share a single element in common? If the cycles α and β share two elements in common? (There are two different answers to this last question!)
Consider the permutation 2 3 4 5)(1 2 3 4 2 3 4 1 5. /(1 3 5 4 2. 1. Express as a product of disjoint cycles. 2. What is the order of the given permutation? 3. Express your answer in number 1 as a product of transposition. 4. Determine whether the permutation is even or odd.
Which of the following is the given permutation expressed as a product of disjoint cycles? * D a = (2678)(124793)(351) in So none of these (26951)(479) (2678)(4935) (248)(35679) (124795)(38) (248)(356)(79) (1248)(35679)
Please answer 1 to 3
(1 374 6) o (2 3 5 4 6) o (8 7 6 2 4 3 5) can be expressed as product of following disjoint cycles (584 3 1) (2 4) (7 6) O (2 4) (7 6) (1 3 5) O (8 4 5) (36 72 1) O None of them (5 8 4) (3 1) (2 4) (7 6)
I. Consider the permutation (1 2 3)(4 5)(1 2 5 4) ~². 1. Express as a product of disjoint cycles. 2. What is the order of the given permutation? 3. Express your answer in number 1 as a product of transpositions. 4. Determine whether the permutation is even or odd.
Suppose that there exists a neighborhood of k > 5 houses, arranged in a circle. The landlords want to connect the houses with roads, according to one of the two following plans: (a) Connect the houses in a circular manner, such that the roads form a single cycle. (b) Construct the complement of plan (a). That is, if two houses had a road between them in plan (a), there is no road between them now. If two houses did not have a road between them in plan (a), there is a road between them now. Prove that there will always be more roads used in plan (b) than in (a).
H. Consider the permutation (1 9 2 3)(1 9 6 5)(1 4 8 7). 1. Express as a product of disjoint cycles. 2. What is the order of the given permutation? 3. Express your answer in number 1 as a product of transpositions. 4. Determine whether the permutation is even or odd. Activat Go to Se

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. Show that a cycle of odd length is an even permutation, and a cycle of even length is an odd permutation. Hence determine which of the following members of S10 are even, or odd: 10) (78) (i) (19 3) ( 6) (45 (ii) (1 9 3 2 6 ) (4 5 10 7 8), (iii) (1 9 3 2 6 4 5 10).