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February 10th, 2015, 10:11 AM   #1
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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.
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February 10th, 2015, 11:04 AM   #2
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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.
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February 10th, 2015, 11:57 AM   #3
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Originally Posted by CRGreathouse View Post
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?
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February 10th, 2015, 12:05 PM   #4
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No, it's saying that

(12)(56)(23)(34)(56) = (12)(23)(34)(56)(56) = (12)(23)(34).
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February 10th, 2015, 12:23 PM   #5
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Originally Posted by CRGreathouse View Post
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.
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February 11th, 2015, 12:08 AM   #6
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They are not “suppressed”. The product of $(5\,6)$ (or any transposition) by itself is the identity.
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February 11th, 2015, 08:24 AM   #7
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Originally Posted by Olinguito View Post
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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