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Problem from Armstrong's book, “Groups and Symmetry”

If $a$, $b$ are members of the permutation group $S_n$, and $ab=ba$. Show that $b$ must be a power of $a$ when $a$ is an $n$-cycle.

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welcome to MathSE. I see that this is your first question. So I wanted to let you know a few things about MathSE. We like to know the sources of questions - if it's homework, please add the [homework] tag. People will still help, so don't worry. We also like to know what you've tried on a problem. These sort of pleasantries usually result in more and better answers. Finally, I should add that posting questions in the imperative (i.e. Compute all such, Prove that...) is considered rude by some of the members, so it would be nice of you to change that wording. Thank you. –  Arturo Magidin Sep 16 '11 at 2:36
    
This comment has been automatically generated, as per the discussion at meta.math.stackexchange.com/q/2968/7850 –  The Chaz 2.0 Sep 16 '11 at 3:27
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marked as duplicate by Arturo Magidin, Srivatsan, Amitesh Datta, anon, Zhen Lin Sep 16 '11 at 2:57

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1 Answer

Recall the following facts.

(1) Two elements of $S_n$ are conjugate if and only if they have the same cycle structure. Hence, the orbit of $a$ under the action of conjugation by elements of $S_n$ is exactly the $n$-cycles.

(2) There are (n-1)! $n$-cycles in $S_n$

(3) The size of the orbit of $a$ under the action of conjugation in $S_n$ times the size of the centralizer of $a$ is equal to the order of $S_n.$

It follows there are exactly $n$ elements that commute with $a.$ As every power of $a$ is such an element, we conclude the $C_{S_n}(a) = \langle a \rangle.$

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Would you mind terribly moving this answer to the other question (I link to it in the comment above)? That one includes a bit more and already has some answers. This is an exact duplicate (down to the wording) of the second half of that answer. –  Arturo Magidin Sep 16 '11 at 2:41
    
Will do. Give me nine key strokes. –  jspecter Sep 16 '11 at 2:46
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If the downvote has to do with the logistics of the repeated question, I would ask the downvoter to remove it. If there is some other reason, kindly explain it. –  Arturo Magidin Sep 16 '11 at 2:47
    
Downvote?? ${}$ –  jspecter Sep 16 '11 at 2:47
    
I know. I hope it's not because of my comment... –  Arturo Magidin Sep 16 '11 at 2:48
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