Relation between $G$-orbits and Cycle Decomposition of a Permutaion.

Question

Let $X_n=\{1,2,...,n\}$, $\delta \in S_n$. Write $G=(\delta)$ and assume $G$ acts on $X_n$. What is the relation between $G$-orbits of $X_n$ and cycle decomposition of $\delta$?

Answer 1

Let $i\in X_n$ and let $k>0$ be minimal such that $\delta^k(i)=i$. Then $(i\delta(i)\cdots\delta^{k-1}(i))$ occurs in the cycle decomposition of $\delta$.

You can argue conversely that if $(a_1a_2\cdots a_k)$ occurs in the cycle decomposition of $\delta$, then $\{a_1,\ldots, a_k\}$ is an orbit of the action. Thus the orbits of the action correspond exactly to the cycles in the cycle decomposition of $\delta$, with the size of the orbit being the length of the cycle and the elements of the cycle being the elements in the orbit.