Short question:
Define \begin{array}{rccc} \rho: & S_3 \times \{1,2,3\} & \rightarrow & \{1,2,3\} \\ & (\sigma,i) & \mapsto & \sigma(i) \end{array} What is $|\text{Orb}(2)|?$ |
The orbit is the possible outputs where $2$ can get sent to if elements of the group act on it. Isn't the orbit in this case not just all of $\{1,2,3 \}$, so $|\text{Orb}(2)|=3$.
This would mean that $|\text{Orb}(i)|=n$ for $i \in \{ 1, 2, \dots n\}$ if we extend this example to $S_n$. Am I getting this correctly?
$\text{Orb}(i)$ is the set of elements that $i$ gets sent to if the group acts on this element, since a group is closed under composition, it containing a cycle generator like $(1 \ 2 \ 3)$ is sufficient (for our example) for every single element to appear in the orbit. Since we then also get $(1 \ 3 \ 2)$ and $(1)(2)(3)$ by repeated composition. The first cycle tells us 3 is an output, the second tells us that 1 is also in there and the identity tells us that 2 can get sent to itself. Therefore $\{1,2,3\}$ is indeed the orbit. This has cardinality $3$
Similarly we can make this case that the cycle $( 1 \ 2 \ 3 \dots n)$ generates an $n$-cycle when it acts upon $S_n$ therefore the orbit will also be all of $\{ 1 \ 2 \ 3 \dots n \}$ .