Abstract Algebra Show that Sn is generated by { (1,2), (1, 2, 3, ...n)} . [Hint Show that as r (1, 2,3, ...)" (1, 2)(1, 2, 3, ..., n)""gives *.... all the transpositions (1,2), (2, 3), (3, 4)... (п — 1, п), (п, 1), Then | show that any transposition is a product of these transpositions and use the following theorem]. Theorem: Any permutation of a finite set containing at least two elements is a product of transpositions. Step 1: Compute the first few transpositions So when, r=0 would get (1,2) when r=1 would get (2,3) when r=2 would get (3,4) if keep going if r=n-1 would get (n,n+1) I am not sure how to obtain the product to obtain (1,n) In essence, going with the pattern above would have a general format being (1, 2, 3, ...,n)" (1,2)(1, 2, 3, ... I know, that n is fixed, so that no matter what n is would get the same transposition as r varies. Just not sure how to obtain (1, n) as not sure what r would be? Step 2: Prove the problem by the use of matematical induction n)"- = (r+ 1,r +2)

Abstract Algebra Show that Sn is generated by { (1,2), (1, 2, 3, ...n)} . [Hint Show that as r (1, 2,3, ...)" (1, 2)(1, 2, 3, ..., n)""gives *.... all the transpositions (1,2), (2, 3), (3, 4)... (п — 1, п), (п, 1), Then | show that any transposition is a product of these transpositions and use the following theorem]. Theorem: Any permutation of a finite set containing at least two elements is a product of transpositions. Step 1: Compute the first few transpositions So when, r=0 would get (1,2) when r=1 would get (2,3) when r=2 would get (3,4) if keep going if r=n-1 would get (n,n+1) I am not sure how to obtain the product to obtain (1,n) In essence, going with the pattern above would have a general format being (1, 2, 3, ...,n)" (1,2)(1, 2, 3, ... I know, that n is fixed, so that no matter what n is would get the same transposition as r varies. Just not sure how to obtain (1, n) as not sure what r would be? Step 2: Prove the problem by the use of matematical induction n)"- = (r+ 1,r +2)

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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