----- > [!proposition] Proposition. ([[kernel iff normal subgroup]]) > For a [[group homomorphism]] $\phi:G \to H$, the set $\text{ker }\phi$ is a [[normal subgroup]] of $G$. > \ > The converse holds too: every [[normal subgroup]] of $G$ is the [[kernel]] of a [[group homomorphism|homomorphism]] — the **natural projection homomorphism** $\pi:G \to G / N$ given by $\pi(a)=aN$.[^1] ^0ce88b > [!proof]- Proof. ([[kernel iff normal subgroup]]) > $\to$. Let $K$ denote the [[kernel of a group homomorphism|kernel]] of $\phi$. We will show it is [[conjugate characterization of normal subgroups|invariant under conjugation]]; let $g \in G$. We have $\phi(gkg^{-1})=\phi(g)\phi(k)\phi(g^{-1})=\phi(g)e_{H}\phi(g ^{-1})=\phi(g g^{-1})=\phi(e)=e_{H},$ > hence $gkg^{-1} \in \ker \phi$ as required. > \ > $\leftarrow.$ Let $N \trianglelefteq G$ contain $n$, then $\pi(n)=nN=N=e_{H}$ so $n \in \ker \pi$. [^1]: "The map which sends each element $a$ to the [[coset]] $aN$ for which it is a representative" ----- #### ---- #### 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 > ```