= Frobenius kernel theorem
{c}
{title2=$N\triangleleft G,\quad G=N\rtimes H$}
For a <Frobenius complement> $H$, a class function $\theta$ with $\theta(1)=0$ satisfies $(\theta^G)_H=\theta$. For each nontrivial $\eta\in\operatorname{Irr}(H)$ of degree $d$, the <virtual character> $(\eta-d1_H)^G+d1_G$ has norm one and positive degree by <Frobenius reciprocity>. It is therefore irreducible, restricts to $\eta$ and is identically $d$ on the candidate kernel $N$. The intersection of the kernels of these characters is exactly $N$: outside $N$, separation follows from the character of the <regular representation> of $H$. This proves normality; $N\cap H=1$ and $|N|=[G:H]$ then give the semidirect product.
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