----- > [!proposition] Proposition. ([[order of quotient group is quotient of orders]]) > Let $G$ be a [[group]] and $H$ a [[normal subgroup]] of $G$. Then, as one might hope, $| G / H |\cdot |H|=|G|.$ > > [!proof]- Proof. ([[order of quotient group is quotient of orders]]) > Straightforward. $|G / H|$ equals the number of [[coset|left cosets]] of $H$ in $G$. [[Lagrange's Theorem|These cosets]] [[partition]] $G$ and are all of [[Lagrange's Theorem#^f23554|the same size]] (that of $H$), thus the [[order of a group|order]] of $G$ equals to the number of cosets times the size of each coset. This is the result. ----- #### ---- #### 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 > ```