Answered: 6. Show that a cycle of odd length is… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.3: Zeros Of Polynomials

Problem 56E

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6. 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:
(i) ( 193 ) ( 26 ) ( 4 5 10 ) (78)
(ii) ( 19326 ) ( 4
5 10 78),
(iii) ( 1932645 10).

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4. (i) Write the following permutations in S9 as a product of disjoint cycles 1 2 3 4 56789 f 7 298 34 165 ) , g= ( 1 2 3 4 5 6 7 8 9 56789 1 2 3 4 (ii) Calculate fg and write it both in two row notation and as a product of disjoint cycles. (iii) Calculate f-1 and write it both in two row notation and as a product of disjoint cycles. (iv) Find the order of f, and compute f20 as a product of disjoint cycles. (v) Determine which of f, g and fg are even or odd.
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: (i) (1 9 3) (2 6) (4 5 10) (78) (ii) (1 9 3 2 6 ) (4 5 10 7 8), (iii) ( 1 9 3 2 6 4 5 10 ).
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