Orbits and stabilizers of the permutation group.

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?

• $G$ is a subset of $S_5$ (subgroup, actually), so why the point-wise definitions of orbit and stabilizer (i.e. $x=1,2,3,4,5$) makes you worry?
– user810157
Sep 20, 2020 at 20:19
• So for example, what are the possible destinations of $1$ when you act on it via an arbitrary element of $G$? (There could conceivably be as many as five, but for this particular group action there are fewer.) And which elements of the entire subgroup $G$ (which has elements in addition to the generators) leave $1$ fixed? Sep 20, 2020 at 20:55

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.
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.