Prove Odd Length Cycle u2 is Also a Cycle

[Metahominid]

Homework Statement

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

Homework Equations

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.

 

[Dick]

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?

 

[Metahominid]

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(a_i) = a_i+1 1\leqi\leqk-1
u(a_k) = a₁

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

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

 

[Dick]

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(a_i) = a_i+1 1\leqi\leqk-1
u(a_k) = a₁

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

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

That's a good enough explanation for me.

 

[Metahominid]

okay, awesome. Thanks