Abstract Algebra Please show using mathematical induction please 6:06Expert Q&Aa product of o and some permutation m An.48. Show that if o is a cycle of odd length, then a is a cycle.49. Following the line of thought opened by Exercise 48, complete the following with a condition involving n andr so that the resulting statement is a theorem:If a is a cycle of length n, then a' is also a cycle if and only if...50, Show that S, is generated by {(1, 2), (1, 2, 3, · , n)}. [Hint: Show that as r varies, (1.2,3, .ny (1.2)(1,2.3. ..ny"-r gives all the transpositions (1, 2), (2, 3), (3,4), .(n – 1,n). (n. 1). Then show that anytransposition is a product of some of these transpositions and use Theorem 8.15.]51. Let a e S. and define a relation on {1, 2, 3, ...,n} by i~j if and only if j =g*(i) for some ke Z.the other. A computation shows that(a), a2, ,an) = (d1,an)(a1, An-1). (a),d3)(d),d2).Therefore any cycle of length n can be written as a product of n-1 transpositions.Since any permutation of a finite set can be written as a product of cycles, we have thefollowing.Any permutation of a finite set containing at least two elements is a product oftranspositions.8.15 Theorem,Naively, this theorem just states that any rearrangement of n objects can be achievedby successively interchanging pairs of them.Abstract Algebra #50 pleaseHomeCoursesTools

Answered: Image /qna-images/answer/c528a0f9-faec-487f-b243-b70e4b8366cc.jpg

Of length I1, tien 50) Show that S is generated by ((1, 2), (1, 2, 3, - · , n)}. [Hint: Show that as r varies, (1.2,3, .ny(1.2) (1.2.3..n"-r gives all the transpositions (1, 2), (2, 3), (3,4),.(n – 1,.n). (n. 1). Then show that any transposition is a product of some of these transpositions and use Theorem 8.15.]

Of length I1, tien 50) Show that S is generated by ((1, 2), (1, 2, 3, - · , n)}. [Hint: Show that as r varies, (1.2,3, .ny(1.2) (1.2.3..n"-r gives all the transpositions (1, 2), (2, 3), (3,4),.(n – 1,.n). (n. 1). Then show that any transposition is a product of some of these transpositions and use Theorem 8.15.]

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 32EQ

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