Order of a m-cycle

**Bernhard** · October 15th 2011, 08:21 PM

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

**alexmahone** · October 15th 2011, 08:43 PM

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 $m-1$ .

**Bernhard** · October 15th 2011, 08:52 PM

Thanks

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

Peter

**alexmahone** · October 15th 2011, 08:54 PM

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 $m-1$ .

**Deveno** · October 15th 2011, 09:01 PM

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

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