Power of an $r$-cycle is an $r$-cycle

Question

I'm trying to prove that if $\alpha$ is an $r$-cycle in $S_n$ then $\alpha^k$ is an $r$-cycle if and only if $(k,r)=1$, but I'm having trouble proving that if $(k,r)=1$ then $\alpha^k$ is an $r$-cycle.

Can someone help me with that?

Answer 1

For the implication you're looking for. Let $\sigma = \alpha ^k$. Then you're trying to show that if $(k,r)=1$, $o(\sigma)=r$. (Where $o(\sigma)$ denotes the order of $\sigma$). Here are the steps:

1. Show $o(\sigma)|r$, the easy bit.
2. Show $r|o(\sigma)$, the hard bit.

For 2. what you want to do let $m$ be such that $\sigma ^m = id$, and then perform the Euclidean division of $m$ by $r$, to get $q, b$ with $0\le b<r$ such that $$m = qr + b$$ and somehow determine that $b=0$. Remember that if $(a,b) = 1$ and $a|bk$ then $a|k$.