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January 27th, 2012, 11:42 PM   #1
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find an inverse of a permutation cycle

Find an inversion of the following cycle



How do we do that??
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January 28th, 2012, 12:58 PM   #2
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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
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January 28th, 2012, 11:48 PM   #3
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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....
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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....
In the link i have provided above...read the 2nd paragraph more carefully...reproduced below for your convenience.

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).


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January 29th, 2012, 12:50 AM   #5
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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
right?
or
right?

and for the original post
ok?
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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
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
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
ok?

(341) is not an inverse, do you understand why?
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