Let $G$ be a group s.t $G = \langle(12),(345)\rangle \subseteq S_5$ acts on the set $X = \{1,2,3,4,5\}$. I want to find all orbits and stabilizers of $G$.
The point I don't understand is that according to the definition of them, they are defined for each element of a set.
Orbit: $$G\cdot x=\{g\cdot x \ \colon g\in G\}$$
Stabilizer: $$G_x=\{g\in G \ \colon g\cdot x=x\}$$
In this case, $G$ permutes multiple elements of $X$ so what are $x$ in this case?
On
Just a few general deductions by using the Orbit-Stabilizer Theorem. Since $G$ has order $6$, in principle the stabilizers might have order $1$ or $2$ or $3$ or $6$ (Lagrange's Theorem); but, $1$ is ruled out because there can't be any orbit of $6$ elements (being $X$ of size $5$), and $6$ is ruled out because there isn't any element of $X$ that is fixed by all the elements of $G$. So, just stabilizers of order $2$ and $3$ are allowed, and accordingly orbits of size $3$ and $2$, respectively. Moreover, the elements of $G$ which move all the $5$ elements of $X$ (namely the products of disjoint $2$- and $3$-cycles), by definition can't be elements of any stabilizer.
The group G is the cyclic group $Z_6$ containing the permutations {(1 2)(345),(354),(12),(345),(12)(354),e}. The orbit 1 is {1,2} and orbit of 3 is {3,4,5}. The stabilizer of 1 is {e,(345),(354)} and for 3 the stabilizer is {e,(12)}. Similarly the orbits and stabilizers of other elements are written down.