---- > [!definition] Definition. ([[stabilizer]]) > Let the [[group]] $G$ [[group action|act on]] the set $X$. The **stabilizer** of an element $x \in X$ is the [[subgroup]] of $G$ $\text{Stab}(x)=\{ g \in G: g(x)=x \}.$ > [!intuition] > The stabilizer of an element is the set of transformations under which it 'is invariant'; under which it 'remains stable'. > [!justification] > Here we verify that for all $x \in X$, $\text{Stab}(x)$ is a [[subgroup]] of $G$. Clearly $e_{G} \in \text{Stab}(x)$ because $e_{G} \cdot x=x$ by the [[group action]] definition. Let $g_{1}, g_{2} \in \text{Stab}(x)$, then $(g_{1}g_{2}) \cdot x = g_{1} \cdot (g_{2} \cdot x)=g_{1} \cdot x = x$ so $g_{1}g_{2} \in \text{Stab}(x)$. Finally, $\text{Stab}(x)$ is closed under inverses because $g_{1} \in \text{Stab}(x) \iff g_{1}x=x \iff x=g^{-1}x$. ![[orbit#^790784]] ---- #### ---- #### 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 > ```