----- > [!proposition] Proposition. ([[subgroups of index 2 are normal]]) > If $H$ is a [[subgroup]] of a [[group]] $G$ with [[index of a subgroup|index]] $2$, then $H$ is a [[normal subgroup]] of $G$. > [!proof]- Proof. ([[subgroups of index 2 are normal]]) > Easy. Recall that from [[Lagrange's Theorem]] and associated **lemmas**, all [[coset]]s of $H$ have equal [[cardinality]] (since $H=eH=He$ is both a left and a right [[coset]] of itself, this [[cardinality]] is $|H|$). $H$ has [[index of a subgroup|index]] 2, meaning that has two disjoint left [[coset]]s and right [[coset]]s: these must be $H \text{ and }G-H$ in both cases. So the left cosets equal the right cosets of $H$ and so $H$ is a [[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 > ```