----- > [!proposition] Proposition. ([[normal subgroup iff intersection of irrep character kernels]]) > Let $G$ be a finite [[group]] and $\{ \rho_{j} \}_{j=1}^{\# \text{ CCs}}$ the finite-dimensional [[irreducible group representation|irreducible]] [[group representation|representations]] of $G$ with corresponding [[character of a representation|characters]] $\{ \chi_{j} \}_{j=1}^{\# \text{CCs}}$. Then a [[subgroup]] $N \leq G$ is [[normal subgroup|normal in]] $G$ if and only if $N$ is an intersection of [[kernel|kernels]] of the $\chi_{j}$ (equivalently, intersection of [[kernel|kernels]] of the $\rho_{j}$). > [!proof]- Proof. ([[normal subgroup iff intersection of irrep character kernels]]) > > Since the [[intersection of subgroups is a subgroup|intersection of normal subgroups is normal]] in $G$ and [[kernel iff normal subgroup]], one direction is clear. > > For the other direction, let $N \trianglelefteq G$. We claim that $N$ is the intersection of the kernels of the lifts $\chi_{1},\dots,\chi_{\ell}$ of all the [[irreducible group representation|irreducibles]] of $G / N$. > > We know that each lift $\chi_{i}$ [[lifting representations|corresponds to]] an [[irreducible group representation]] of $G$ whose [[kernel]] contains $N$. So certainly $N \leq \bigcap_{i=1}^{\ell} \ker \chi_{\ell}$. Now let $g \in G / N$ with $gN \neq N$, hence (using notation as in [[lifting representations]]) $\tilde{\chi}(gN) \neq \tilde{\chi}(N)$ for *some* irreducible $\tilde{\chi}$ of $G / N$. Lifting $\tilde{\chi}$ to $\chi$, [[lifting representations|we know that]] $\chi(g)=\tilde{\chi}(gN) \neq \tilde{\chi}(N)=\chi(1)$. So $\chi(g) \neq \chi(1)$, meaning that corresponding $\rho(g) \neq I$ and hence $g \notin \ker \rho= \ker \chi$. for some $\chi \in \{ \chi_{1},\dots, \chi_{\ell} \}$. So if $g \notin N$ then $g \notin \bigcap_{i=1}^{\ell} \ker \chi_{\ell}$, and the contrapositive of this yields the reverse inclusion. ----- #### ---- #### 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 > ```