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

In summary, a k-cycle in a permutation of length n has an order of k, meaning that when it is raised to the power of k, it returns to the identity permutation. This can be seen by considering special cases and geometric interpretations.
  • #1
AxiomOfChoice
533
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?
 
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  • #2
That is surely obvious, isn't it?
 
  • #3
matt grime said:
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
Special cases are always a good way to get started. Try k=1,2,3.

P.S. you meant "least positive integer"
 
  • #5
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
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?
 

1. What does it mean for x to be a cycle of length n?

A cycle of length n means that x is a permutation that consists of n distinct elements and when applied repeatedly, it will eventually return to its original form after n steps.

2. What is the significance of x^n being the identity?

The identity is a permutation that leaves all elements unchanged when applied. Therefore, if x is a cycle of length n, x^n being the identity means that after n repetitions, the elements will return to their original positions and the cycle is complete.

3. Can x be a cycle of any length?

Yes, x can be a cycle of any length as long as it consists of n distinct elements and x^n is the identity.

4. How is this concept related to group theory?

This concept is related to group theory because it falls under the category of permutations, which is a fundamental concept in group theory. The set of all permutations of n elements forms a group under the operation of composition.

5. Can this concept be applied to any mathematical structure other than groups?

Yes, this concept can be applied to other mathematical structures such as rings, fields, and vector spaces. In these structures, the operation of multiplication can be represented as repeated addition or composition, making the concept of x^n being the identity applicable.

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