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The following statement provides (if true) a powerful tool for inductive proofs. Can anyone confirm if it is true:

Suppose $G$ is a finite group and $N$ a normal subgroup of $G$. Is it true that if $\chi\in Irr(G/N)$ is a monomial character of $G/N$ of degree $\chi(1)$, then $G$ has a monomial character of same degree?

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    $\begingroup$ Can't we just take the inflation of $\chi$ to $G$? It is still monomial, because inflation commutes with induction. I.e.: if $\chi = \phi\!\uparrow_{K/N}^{G/N}$ is an irreducible monomial character, then $(\mathrm{Inf}^K \phi) \!\uparrow_K^G = \mathrm{Inf}^G (\phi\!\uparrow_{K/N}^{G/N}) = \mathrm{Inf}^G \chi$ is an irreducible monomial character of $G$. $\endgroup$
    – Mark Wildon
    Commented Jan 8, 2019 at 2:58
  • $\begingroup$ Ah yes, I think actually this suffices! $\endgroup$
    – user129425
    Commented Jan 8, 2019 at 9:47

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