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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. I). Then show that any transposition is a product of some of these transpositions and use Theorem 8.15.]

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. I). Then show that any transposition is a product of some of these transpositions and use Theorem 8.15.]

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
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Abstract Algebra 

7:33 M NA 5GÈ | Expert Q&A a product ofo and some permutation in 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 and r so that the resulting statement is a theorem: If o 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 any transposition 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 the following. 8.15 Theorem/ Any permutation of a finite set containing at least two elements is a product of transpositions. Naively, this theorem just states that any rearrangement of n objects can be achieved by successively interchanging pairs of them. Abstract Algebra #50 please 88 Home Courses Tools
Transcribed Image Text:7:33 M NA 5GÈ | Expert Q&A a product ofo and some permutation in 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 and r so that the resulting statement is a theorem: If o 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 any transposition 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 the following. 8.15 Theorem/ Any permutation of a finite set containing at least two elements is a product of transpositions. Naively, this theorem just states that any rearrangement of n objects can be achieved by successively interchanging pairs of them. Abstract Algebra #50 please 88 Home Courses Tools
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