If x is a cycle of length n, x^n is the identity.

  • #1
Is it true that if [itex]\sigma \in S_n[/itex] is a cycle of length [itex]k \leq n[/itex], then [itex]\sigma^k = \varepsilon[/itex], where [itex]\varepsilon[/itex] is the identity permutation, and that [itex]k[/itex] is the least nonzero integer having this property?
 

Answers and Replies

  • #2
matt grime
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That is surely obvious, isn't it?
 
  • #3
That is surely obvious, isn't it?
Not to me. :frown: Maybe I'm missing something small...if you can get me started on why it's the case, I can probably finish it out.
 
  • #4
Hurkyl
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Special cases are always a good way to get started. Try k=1,2,3.

P.S. you meant "least positive integer"
 
  • #5
matt grime
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A k-cycle has order k - it really is trivial. You only need to consider the case of

(123..k)

which just rotates the elements 1,..,k cyclically.
 
  • #6
matt grime
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More geometrically, label the vertices of a k-gon with 1,..,k, then (1...k) rotates it by 2pi/k.

If you don't like that then just think what (1...k) does to the set 1,..,k it sends i to i+1 (wrapping k round to 1). So what happens if apply it r times?
 

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