Powers of $m$-cycles are also $m$-cycles 
Let $\sigma$ be the $m$-cycle $(1~2~...~m)$. Show that $\sigma^i$ is also an $m-$cycle if and only if $i$ is relatively prime to $m$. 

I having some difficulty with this problem. Through experimenting with some simple cycles, I have seen a cycle "split" into a product of disjoint cycles when its length is not relatively prime to $i$, but I have no way of rigorously describing this "splitting." I guess my question is, how do distinguish between cycles and "split" cycles; and how could this help with solving the problem?

An $\;m\,-$ cycle $\;\sigma\;$ is an element of order $\;m\;$ in the group $\;S_m\;$ and as in any other group, we have that $\;\langle \sigma^k\rangle=\langle\sigma\rangle\iff (m,k)=1\;$ , so one direction is immediate...and the other one almost: if $\;(k,m)=d>1\;$ , then $\;\sigma^k\;$ has order less than $\;m\;$ so it can not be an $\;m\,-$ cycle.
