= Solution
A finite <monomial group>, also called an <M-group>, is a <finite group> for which every <irreducible character> is a <monomial character>: for every $\chi\in\operatorname{Irr}(G)$ there are a <subgroup> $H\leq G$ and a <linear character> $\lambda$ of $H$ such that $\chi=\operatorname{Ind}_H^G\lambda$. The inducing <subgroup> and <linear character> may depend on $\chi$.
<Taketa's theorem> states that \b[every finite monomial group is solvable]. Here being a <solvable group> means that the iterated <derived subgroups> eventually reach the identity.
<Dade's embedding theorem for monomial groups> states that \b[every finite solvable group embeds as a <subgroup> of a finite monomial group]. It does not say that every solvable group is itself monomial, nor that the embedding can always be taken normal. Together the results identify solvability as a necessary property of monomial groups without making monomiality closed under taking arbitrary <subgroups>.
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