Problem on Symmetric Groups

Question

Prove that a cycle of length $L = k · m$, taken to the $k^{\text{th}}$ power, will decompose into $k$ cycles, each of length $m$.

I have no idea on this one. Kindly help.

Thanks in advance.

Answer 1

Let us take the cycle $\sigma=(1,2,...,mk)$. Then $$\sigma(a)=a+1$$ $$\sigma^2(a)=a+2$$
...
$$\sigma^k(a)=a+k$$ (all done modulo $km$). Thus, $$\sigma^k=(1,k+1,...,(m-1)k+1)(2,k+2,...(m-1)k+2)...(k-1,2k-1,...,mk-1)$$

and those are $k$ cycles of length $m$.