Prove Odd Length Cycle u2 is Also a Cycle

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Homework Statement


Show that for any cycle of odd length, u, then u2 is a cycle as well.


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The Attempt at a Solution


I looked at examples of it and it does work them but I am unsure of how to generalize it.
 
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There is really only one three cycle, (1,2,3). All of the others just relabel the elements. Ditto for other lengths. Are you sure you can't figure out why if the length of the cycle is odd then a^2 is a cycle?
 
I think I was just really tired and didn't remember what I had looked at before.

but my proof was if u is a cycle of length k.
Then
u(ai) = ai+1 1\leqi\leqk-1
u(ak) = a1

then if v = u2
v(ai) = ai+2 1\leqi\leqk-2
v(ak-1) = a1
v(ak) = a2

so v(a1) = a3 which then we permute, and each goes on to an odd a, until k-2 which goes to ak, this goes to a2 which then covers the even a's, and at ak-1 goes back to a1 making another cycle.
 
Metahominid said:
I think I was just really tired and didn't remember what I had looked at before.

but my proof was if u is a cycle of length k.
Then
u(ai) = ai+1 1\leqi\leqk-1
u(ak) = a1

then if v = u2
v(ai) = ai+2 1\leqi\leqk-2
v(ak-1) = a1
v(ak) = a2

so v(a1) = a3 which then we permute, and each goes on to an odd a, until k-2 which goes to ak, this goes to a2 which then covers the even a's, and at ak-1 goes back to a1 making another cycle.

That's a good enough explanation for me.
 
okay, awesome. Thanks
 
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