---- > [!definition] Definition. ([[kernel of a group homomorphism]]) > The **kernel** of a [[group homomorphism]] $\phi:G \to H$ is the [[subgroup]] $\text{ker }\phi := \{ g \in G : \phi(g) = e_{H} \}=\phi ^{-1}(e_{H}).$ > > $\ker \phi$ is 'kernel' in the [[categorical kernel|categorical sense]]: see [[universal property of group homomorphism kernels]]. > [!justification] > We need to show $\text{ker }\phi$ is indeed a [[subgroup]] of $G$. Clearly $e_{G} \in \text{ker }\phi$. If $g, h \in \text{ker }\phi$, then $\phi(gh)=\phi(g)\phi(h)=e_{H}e_{H}=e_{H}$, so $\text{ker }\phi$ is closed under the [[binary operation|group operation]]. Finally, if $g \in \ker \phi$, then we have $\phi(g g^{-1})=e_{H}=\phi(g)\phi(g^{-1})=e_{H}g ^{-1}$, so closure under inverses holds. > [!basicproperties] > - [[kernel iff normal subgroup]] ---- #### ---- #### 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 > ```