----- > [!proposition] Proposition. ([[character table determines normal subgroups]]) > The [[character of a representation|character]] [[character table|table]] of a finite-dimensional [[group representation|representation]] $(\rho, V)$ on a finite group $G$ tells us all of the [[normal subgroup|normal subgroups]] of $G$. > \ > To see this, note first that any [[character table]] contains characters corresponding to all [[irreducible group representation|irreps]], and thus all 1-dimensional [[group representation|representations]] since these are necessarily [[irreducible group representation|irreducible]]. [[character of a representation#^b39c84|1D characters are]] [[group homomorphism|homomorphisms]] $G \xrightarrow{\chi} \mathbb{C}^{\times}$ which, since $\chi$ is a [[class function]], have as [[kernel|kernels]] unions of [[conjugate|conjugacy classes]] of $G$. Then because [[normal subgroup iff intersection of 1D character kernels]] > [!proof]- Proof. ([[character table determines normal subgroups]]) > ~ ----- #### ---- #### 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 > ```