---- > [!definition] Definition. ([[commutator]]) > Let $G$ be a [[group]] and $x,y \in G$. The **commutator** of $x$ and $y$ is the element $xyx ^{-1}y^{-1} \in G$. It is denoted $[x,y]$. > \ > **Remark.** $[x,y]$ is trivial if and only if $x$ and $y$ [[commute]]. > > Let $R$ be a [[ring]] and $a,b \in R$. The **commutator** of $a$ and $b$ is the element $ab-ba$. ---- #### ---- #### 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 > ```