Having defined group actions and G-Sets, the next natural question is what happens to a single element when every group element is allowed to act on it. The collection of all positions reachable from that element is called its orbit, while the group elements that leave it unchanged form its stabilizer. These two ideas are the first structural tools in group actions because they separate movement from symmetry. The focus keyword for this lesson is orbits and stabilizers. In this class note, we shall prove that orbits form equivalence classes, define stabilizers as subgroups, and compute them in the natural action of $S_3$.

Let $G$ be a group and let $S$ be a $G$-set. Define a relation $\sim$ on $S$ by $$\begin{equation*} a\sim b \end{equation*}$$ if and only if there exists $g\in G$ such that $$\begin{equation*} g\cdot a=b. \end{equation*}$$ This relation says that $b$ is reachable from $a$ by the action of some element of $G$.