Proof that in a cyclic group an element of order d is a product of $\frac{n}{d}$d-cycles

Question

I have to Proof that in a cyclic group of order n an element of order d is a product of $\frac{n}{d}$d-cycles. I know that the order of an element is d = $\frac{n}{gcd(n,m)}$. How can I conclude my thereom? Is there any proof for that?

Answer 1

If you mean that in the Cayley representation of $G$, an element $g$ of order $d$ is a product of $\frac{n}{d}$ $d$-cycles, then this is clear, because the cycles are orbits of $x \mapsto gx$ and these are the cosets of the $\langle g \rangle$. The number of cosets is the index of $\langle g \rangle$, which is $\frac{n}{d}$.