Let $S_A$ be the group of permutations of a set $A$ (finite or infinite). We say that a permutation $\sigma\in S_A$ is a finite cycle iff $\sigma=(a_1,a_2,\dots,a_n)$ where $a_i\in A$ for $1\leq i\leq n$. $i,e$ $\sigma(a_i)=a_{i+1}$ for $i<n$, $\sigma(a_n)=a_1$ and $\sigma(b)=b$ for each $b\in A\setminus\{a_1,\dots, a_n\}$.
I prove the following statement: Let $r,n\in\mathbb{N}$ and $\sigma\in S_A$ a finite cycle with length $n$. If $n$ is odd and $r=2^k$ for some $k>0$ then $\sigma^r$ is a cycle.
My question: The previous statement is a "if, and only if" statement?