---- > [!definition] Definition. ([[orbit]]) > Fix a [[group action]] of a [[group]] $G$ on a set $X$. The **orbit** of an element $x \in X$ is the subset of $X$ $O(x):=\{ g \cdot x : g \in G \}=.$ > [!intuition] > So-called because it represents all the 'places which $x$ can be taken by $G, like Earth's orbit represents 'all the places it can be taken by gravity'. > \ > Literal [[integral curve|orbits]] around [[center of mass|centroids]] in $M_{2}$. > [!basicexample] Consider the [[group action]] [[group action#^363027|given by conjugation]], $\begin{align} G \times G & \to G \\ (g,x) & \mapsto gxg^{-1}. \end{align}$ The [[orbit]] of $x \in X$ is the [[conjugate|conjugacy class]] of $x$: $O(x)=\{ gxg^{-1} : x \in X \}=[x].$ The [[stabilizer]] of $x \in X$ is the [[centralizer of an element in a group|centralizer]] of $x$ in $G$: $\text{Stab}(x)=\{ g \in G : gxg^{-1}=x \}=\{ g \in G: gx=xg \}=Z_{G}(x).$ ^790784 > [!justification] > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```