My Math Forum find an inverse of a permutation cycle

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 January 27th, 2012, 11:42 PM #1 Senior Member   Joined: Sep 2010 From: Germany Posts: 153 Thanks: 0 find an inverse of a permutation cycle Find an inversion of the following cycle $(143)$ $(143)^{-1}$ How do we do that??
 January 28th, 2012, 12:58 PM #2 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234 Re: find an inverse of a permutation cycle just write the numbers in reverse order, so the inverse is (341) http://www.math.csusb.edu/notes/advance ... node9.html
 January 28th, 2012, 11:48 PM #3 Senior Member   Joined: Sep 2010 From: Germany Posts: 153 Thanks: 0 Re: find an inverse of a permutation cycle that is very strange because for example in my book the author gives such examples (123) has an inverse (132) and (132) has an inverse (123) and (135) has (153) so it seems that the only numbers that change their places are the middle one and the last one....
January 29th, 2012, 12:33 AM   #4
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Re: find an inverse of a permutation cycle

Quote:
 Originally Posted by rayman that is very strange because for example in my book the author gives such examples (123) has an inverse (132) and (132) has an inverse (123) and (135) has (153) so it seems that the only numbers that change their places are the middle one and the last one....

To find the inverse of a permutation that is a cycle all we have to do is write the elements of the cycle in reverse order. Thus the inverse of (1 2 3 4) is (4 3 2 1). Since a cycle can be written with any of its elements as the first term we can also write this inverse as (1 4 3 2). This gives an alternative way to write down the inverse of a cycle. Fix the first element in the cycle and write the remaining elements in reverse order. Thus, the inverse of (1 2 3 4 5) is (1 5 4 3 2).

 January 29th, 2012, 12:50 AM #5 Senior Member   Joined: Sep 2010 From: Germany Posts: 153 Thanks: 0 Re: find an inverse of a permutation cycle ah okej, finally I got it So it does not matter which order I take then for ex $(123)^{-1}=(132)=(321)$ right? or $(1234)^{-1}=(4321)=(1324)$ right? and for the original post $(134)^{-1}=(143)=(341)=(431)$ ok?
January 29th, 2012, 01:26 AM   #6
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Re: find an inverse of a permutation cycle

Quote:
 Originally Posted by rayman ah okej, finally I got it So it does not matter which order I take then for ex $(123)^{-1}=(132)=(321)$ right? For this example you are correct, notice (132) contains the same information as (321) Both representations send 1 to 3, 3 to 2, and 2 to 1 So, it may appear that these 2 cycles are different inverses but THEY ARE NOT! They are the same inverse. In a group the inverse must be UNIQUE, and permutation cycles form a group. or $(1234)^{-1}=(4321)=(1324)$ right? For this example you are not entirely correct because the representations (4321) and (1324) do not contain the same information so they are not the same unique inverse. For instance in the 1st one 4 goes to 3 but in the other 4 goes to 1 (4321) is the inverse so is (1432) THEY ARE BOTH THE SAME INVERSE CAUSE THEY REPRESENT THE SAME INFORMATION. you cannot scramble the numbers arbitrarily. and for the original post $(134)^{-1}=(143)=(341)=(431)$ ok? (341) is not an inverse, do you understand why?

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