---- > [!definition] Definition. ([[centralizer of an element in a group]]) > Let $G$ be a [[group]] and $x \in G$. The **centralizer of $x$ in $G$**, denoted $Z_{G}(x)$, is the [[subgroup]] $Z_{G}(x):=\{ g \in G : gx=xg \}= \{ g \in G : x = gxg^{-1} \}.$ > [!justification] > We must show $Z_{G}(x)$ is a [[subgroup]] of $G$. Clearly $e \in Z_{G}(x)$ because $ex=x=xe$. Let $g_{1},g_{2} \in Z_{G}(x)$. Then $(g_{1}g_{2})(x)=g_{1}(g_{2}x)=g_{1}xg_{2}=x(g_{1}g_{2}),$ > hence $g_{1}g_{2} \in Z_{G}(x)$ as well. Finally, for $g \in Z_{G}(x)$, we have that $gx=xg \implies x=gxg^{-1} \implies g^{-1}x=xg^{-1}$ > and so closure under inverses holds. ---- #### ---- #### 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 > ```