# Thread: Order of a m-cycle

1. ## Order of a m-cycle

Dummit and Foote Section 1.3 Symmetric Groups - Exercise 10 states:

Prove that if $\sigma$ is the m-cycle ( $a_1$ $a_2$ ... $a_m$) then for all i $\in$ {1,2, ... m} we have ${\sigma}^i$( $a_k$) = $a_{k+i}$ where k+i is replaced by its least positive residue mod m. Deduce that | $\sigma$| = m.

Can anyone help with this problem? Would be grateful for help!

I think I can see how this works but I am struggling with how to write a clear and valid formal proof.

Peter

2. ## Re: Order of a m-cycle

Originally Posted by Bernhard
Dummit and Foote Section 1.3 Symmetric Groups - Exercise 10 states:

Prove that if $\sigma$ is the m-cycle ( $a_1$ $a_2$ ... $a_m$) then for all i $\in$ {1,2, ... m} we have ${\sigma}^i$( $a_k$) = $a_{k+i}$ where k+i is replaced by its least positive residue mod m. Deduce that | $\sigma$| = m.

Can anyone help with this problem? Would be grateful for help!

I think I can see how this works but I am struggling with how to write a clear and valid formal proof.

Peter
Note that $\sigma^m=e$ and $\sigma^n\neq e$ for $n$ ranging from $1$ to $m-1$.

3. ## Re: Order of a m-cycle

Thanks

Yes, can see that! Would that constitute a satisfactory proof?

Peter

4. ## Re: Order of a m-cycle

Originally Posted by Bernhard
Would that constitute a satisfactory proof?
Depends on whether you have a satisfactory argument for $\sigma^n\neq e$ for $n$ ranging from $1$ to $m-1$.

5. ## Re: Order of a m-cycle

if you prove the first part, you'll have a satisfactory proof of the second part.