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 Post subject: Powers of Permutations.Posted: Tue, 21 Feb 2012 01:33:54 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
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I'm doing #3. sovalpha is of length s and its a cycle, alpha=(a1 a2....aS). so alpha inverse=(aS...a2 a1).
And now its saying, alpha inverse=alpha^(S-1) Now maybe im misunderstanding, but for example lets use Beta is a cycle of length 5 so B=(b1 b2 b3 b4 b5), does this mean that the inverse of Beta is also equal to (b1 b2 b3 b4) ,since its 5-1=4?

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 Post subject: Re: Powers of Permutations.Posted: Tue, 21 Feb 2012 01:58:13 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
DgrayMan wrote:

I'm doing #3. sovalpha is of length s and its a cycle, alpha=(a1 a2....aS). so alpha inverse=(aS...a2 a1).
And now its saying, alpha inverse=alpha^(S-1) Now maybe im misunderstanding, but for example lets use Beta is a cycle of length 5 so B=(b1 b2 b3 b4 b5), does this mean that the inverse of Beta is also equal to (b1 b2 b3 b4) ,since its 5-1=4?

No, one has order 5 the other has order 4, just because you raise something to a certain power doesn't mean you lose what things go to, if you look at the original permutation:

by definition this means skip things and go to the next one, you should be able to see that this means the same thing as "go to the previous thing in the cycle listed" i.e. the inverse is , since cycles are the same up to cyclic permutation.

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 Post subject: Re: Powers of Permutations.Posted: Tue, 21 Feb 2012 03:52:50 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
Posts: 182
so a^(s-1)= (a1 a3... as-2 as) ?

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 Post subject: Re: Powers of Permutations.Posted: Tue, 21 Feb 2012 03:57:03 UTC
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DgrayMan wrote:
so a^(s-1)= (a1 a3... as-2 as) ?

No, it's what I wrote in my last post, and what you said you suspected it was in your first post.

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 Post subject: Re: Powers of Permutations.Posted: Tue, 21 Feb 2012 04:29:43 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
Posts: 182
hmm. Is there somewhere where i can find examples. im still confused?

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 Post subject: Re: Powers of Permutations.Posted: Tue, 21 Feb 2012 05:02:54 UTC
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DgrayMan wrote:
hmm. Is there somewhere where i can find examples. im still confused?

Examples of what?

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 Post subject: Re: Powers of Permutations.Posted: Tue, 21 Feb 2012 06:05:17 UTC
 S.O.S. Oldtimer

Joined: Sat, 21 Jan 2012 03:59:22 UTC
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Well what about a permutation of length s= 5 . B=(1 2 3 4 5), and its inverse B^(-1)=(5 4 3 2 1)
what would B^(s-1) look like?

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 Post subject: Re: Powers of Permutations.Posted: Tue, 21 Feb 2012 07:46:37 UTC
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DgrayMan wrote:
Well what about a permutation of length s= 5 . B=(1 2 3 4 5), and its inverse B^(-1)=(5 4 3 2 1)
what would B^(s-1) look like?

It would look like the same thing, we already established that for an s-cycle , , and inverses in a group are unique, so they must be the same thing.

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 Post subject: Re: Powers of Permutations.Posted: Tue, 21 Feb 2012 14:43:35 UTC
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DgrayMan wrote:
so a^(s-1)= (a1 a3... as-2 as) ?

No, it's what I wrote in my last post, and what you said you suspected it was in your first post.

In fact, is not a power of a unless s=2.

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