---- > [!theorem] Theorem. ([[the correspondence theorem]]) > Suppose that $N$ is a [[normal subgroup]] of a [[group]] $G$, and let $\phi:G \to G / N$ denote the [[kernel iff normal subgroup|natural projection homomorphism]]. > 1. There is a [[bijection]] $\{\text{subgroups of $G$ containing N}\} \leftrightarrow \{ \text{subgroups of $G / N$} \}$ > given by $H \mapsto \phi(H)=H / N$ > for $H$ a [[subgroup]] of $G$, with the inverse map given by $\overline{H} \mapsto \phi ^{-1}(\overline{H})$ > for $\overline{H}$ a [[subgroup]] of $G / N$. *Moreover*, in this [[bijection]] [[normal subgroup]]s correspond to [[normal subgroup]]s. > [!proof]- Proof. ([[the correspondence theorem]]) > ![[third isomorphism theorem#1]] ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as 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 > ```