= Abelian intersection of irreducible inducing subgroups
{title2=$K=\bigcap\{H:\operatorname{Ind}_H^G\varphi\text{ is irreducible for some }\varphi\}$}
In a finite <monomial group>, the displayed intersection is a <normal subgroup>. Every irreducible character is induced from a linear character of a subgroup containing $K$. On restriction to $K$, all conjugate inducing lines remain invariant, so $K'$ lies in every irreducible character kernel. The intersection of those kernels is trivial, proving that $K$ is <Abelian>.
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