= Frobenius kernel criterion by irreducible induction
{c}
{title2=$\operatorname{Ind}_N^G\varphi\in\operatorname{Irr}(G)\quad(1_N\ne\varphi\in\operatorname{Irr}(N))$}
For $1<N\triangleleft G$ proper, the displayed condition is equivalent to $N$ being a <Frobenius kernel>. For a normal subgroup the induction norm is $|I_G(\varphi):N|$. Trivial inertia for each nonprincipal character and <Brauer's permutation lemma> imply that every element outside $N$ has trivial centralizer in $N$. This forces $N$ to be a normal Hall subgroup; a complement is then fixed-point-free on $N$.
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