Regular permutations $ \sigma \in  S_n $ is a regular permutation if there exist disjoint cycles $ \tau_1...\tau_r $ of length m such that $ \sigma = \tau_1 \circ ... \circ \tau_r $ and $supp(\tau_1)\cup...\cup supp(\tau_r) = \{1...n\}$
I have been asked to prove that:
$\sigma $ is a regular permutation $\Longleftrightarrow$ $ \exists $ some $k$ natural and some cycle $\tau \in S_n $ such that $\sigma = \tau^k$.
I have observed that $ n =rm$ , and that the orbit of $\sigma$ is $\{1...n\}$ , but I just cannot figure out how to continue. Can someone give me some tips? 
Thank you in advance. 

Interleave the entries from each cycle to create one big cycle that starts with the first element of each cycle then (in the same ordering of cycles) the second element of each cycle, then the third element of each cycle, etc.
