My Math Forum Permutation being written as product of cycles.
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 February 10th, 2015, 10:11 AM #1 Member   Joined: Jun 2012 From: San Antonio, TX Posts: 84 Thanks: 3 Math Focus: Differential Equations, Mathematical Modeling, and Dynamical Systems Permutation being written as product of cycles. Course: Abstract Algebra I need help understanding the following fact from the page http://www.math.clemson.edu/~kevja/C...ection-4.1.pdf Fact: Any permutation $\sigma \in S_n$ can be written as a product of transpositions (2-cycles). The page says that (12)(23)(34) = (12)(56)(23)(34)(56). I can't see how they are equal.
 February 10th, 2015, 11:04 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Well, you could go through each element one-by-one and see where it's mapped. Or else you could notice that (56) only affects those two elements, and the others don't affect them, so you can split into those cases which makes the result pretty evident. Thanks from MadSoulz
February 10th, 2015, 11:57 AM   #3
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Quote:
 Originally Posted by CRGreathouse Well, you could go through each element one-by-one and see where it's mapped. Or else you could notice that (56) only affects those two elements, and the others don't affect them, so you can split into those cases which makes the result pretty evident.
I see. The (56) just appeared out of "nowhere" for me.

With respect to the fact I have posted, all it is saying is that I could write permutation $\sigma$= (1234)(56) as (12)(23)(34)(56), correct?

 February 10th, 2015, 12:05 PM #4 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms No, it's saying that (12)(56)(23)(34)(56) = (12)(23)(34)(56)(56) = (12)(23)(34).
February 10th, 2015, 12:23 PM   #5
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Quote:
 Originally Posted by CRGreathouse No, it's saying that (12)(56)(23)(34)(56) = (12)(23)(34)(56)(56) = (12)(23)(34).
(56) is a 1-cycle so why are they suppressed? Does it have something to do with the identity?

*EDIT Should read as:
(56) isn't a 1-cycle so why are they suppressed? Does it have something to do with the identity?

Last edited by MadSoulz; February 10th, 2015 at 12:29 PM.

 February 11th, 2015, 12:08 AM #6 Senior Member     Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra They are not “suppressed”. The product of $(5\,6)$ (or any transposition) by itself is the identity. Thanks from MadSoulz
February 11th, 2015, 08:24 AM   #7
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Quote:
 Originally Posted by Olinguito They are not “suppressed”. The product of $(5\,6)$ (or any transposition) by itself is the identity.
I see. Thank you for clearing that up.

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