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,...
Related questions
Question
![7:33 AT&T
NA X 56E al-
E Expert Q&A
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, ..., n)"-"
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
— (r+1,r+2)
what r would be?
Step 2:
Prove the problem by the use of matematical
induction](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47472cc3-819f-4272-92fe-e39219493627%2F8068c924-6925-483e-a22a-8999eb996191%2Frd0vutm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:7:33 AT&T
NA X 56E al-
E Expert Q&A
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, ..., n)"-"
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
— (r+1,r+2)
what r would be?
Step 2:
Prove the problem by the use of matematical
induction
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