My Math Forum PERMUTATION GROUPS

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February 14th, 2012, 08:31 AM   #31
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Re: PERMUTATION GROUPS

Quote:
 Originally Posted by agentredlum How do i write this permutation using your notation? $\begin{pmatrix} 1 &2 & 3 &4 &5 \\ 5 &1 &4 &3 &2 \end{pmatrix}$

Open parenthesis (
Start with the smallest number, and "track it" until you get back to that number. Close parenthesis)
Then track the next smallest number (that isn't already) in a cycle.
Eventually you will have all numbers.

e.g. we track 1 --> 5 since s(1) = 5. Then we follow 5 --> 2 since s(5) = 2. Then we follow 2 --> 1, since s(2) = 1. Arrived back at 1, we stop and write
(1 5 2).
The smallest number not in this cycle is 3.
s(3) = 4 and s(4) = 3, so we write
(34)

In conclusion, (1 5 2)(3 4).

 February 14th, 2012, 08:39 AM #32 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: PERMUTATION GROUPS I see... so it is impossible to write it using 1 set of parenthesis, or as you would say, a product of 1 cycle. Well, i'm glad we got that cleared up!
 February 14th, 2012, 08:42 AM #33 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4 Re: PERMUTATION GROUPS Right. In a proper cycle, everything is necessarily moved, where that need it be the case for arbitrary bijections/permutations.
 February 14th, 2012, 08:52 AM #34 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: PERMUTATION GROUPS So the meaning of (5 1 4 3 2) in your notation is $\begin{pmatrix} 1 &2 & 3 &4 &5 \\ 4 &5 &2 &3 &1 \end{pmatrix}$ in my notation.
 February 14th, 2012, 08:57 AM #35 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4 Re: PERMUTATION GROUPS yes!
 February 14th, 2012, 09:16 AM #36 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: PERMUTATION GROUPS Thanx Chaz! For the original problem, i converted everything to my notation, got the right answer my way, then converted to your notation and posted. Doing it your way is much faster and well worth the effort to understand your way. I'm glad we had this conversation

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