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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...ection4.1.pdf Fact: Any permutation $\sigma \in S_n$ can be written as a product of transpositions (2cycles). 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 onebyone 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.

February 10th, 2015, 11:57 AM  #3  
Member Joined: Jun 2012 From: San Antonio, TX Posts: 84 Thanks: 3 Math Focus: Differential Equations, Mathematical Modeling, and Dynamical Systems  Quote:
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  
Member Joined: Jun 2012 From: San Antonio, TX Posts: 84 Thanks: 3 Math Focus: Differential Equations, Mathematical Modeling, and Dynamical Systems  Quote:
*EDIT Should read as: (56) isn't a 1cycle 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.

February 11th, 2015, 08:24 AM  #7 
Member Joined: Jun 2012 From: San Antonio, TX Posts: 84 Thanks: 3 Math Focus: Differential Equations, Mathematical Modeling, and Dynamical Systems  

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