What is the orbit of a permutation?

Question

I have the following statement: $\sigma, \tau \in S_n$ are conjugated if and only if their orbits have the same length. I only know orbits in the context of groups acting on a set. Namely, let $X$ be a set and $G$ be a group. Then, $Gx := \{gx |\, g \in G\}$ is called orbit of $x$ in $G$. What in this case is $X$ and what is $G$?

Answer 1

The symmetric group $S_n$ is defined via its action on the set $X=\{1,2,\ldots,n\}$, so unless otherwise specifed (i.e. unless action on $G$ itself by conjugation or left/reight multiplication or the action on some other set such is expressly mentioned) one may assume that this action is being referred to. The orbit of an element $x\in X$ is apparently simply the set of points in the cycle containing $x$. So for example in $S_7$, the permutation $\sigma =(1\;3)(2\;6\;5)$ has one orbit of length $2$ (namely $\{1,3\}$), one of length $3$ (namely $\{2,5,6\}$) and two orbits of length $1$ (namely $\{4\}$ and $\{7\}$). You will notice that $\tau=(7\;6)(5\;4\;3)$ has exactly the same orbit lengths as $\sigma$. And indeed a bijection $\pi\colon X\to X$ (which just says that $\pi\in S_7$ as well) that respects these orbits suitably, such as $\pi(1)=7$, $\pi(3)=6$, $\pi(2)=5$, $\pi(6)=4$, $\pi(5)=3$, $\pi(4)=2$, $\pi(7)=1$, shows us that $\sigma,\tau$ are conjugate: $\tau\circ \pi=\pi\circ\sigma$.