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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 $K^{\prime }$ lies in every irreducible character kernel. The intersection of those kernels is trivial, proving that $K$ is Abelian.

Every finite solvable group is isomorphic to a subgroup of a finite monomial group. This is an embedding assertion and does not assert that every solvable group is monomial, nor that the embedding is normal.

A finite monomial group, also called an M-group, is a finite group for which every irreducible character is a monomial character: for every $\chi \in \mathrm{Irr}(G)$ there are a subgroup $H\le G$ and a linear character $\lambda$ of $H$ such that $\chi ={\mathrm{Ind}}_H^G\lambda$. The inducing subgroup and linear character may depend on $\chi$.

Taketa's theorem states that 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 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.

Every finite monomial group is a solvable group. Thus expressing every irreducible character as induction from a linear character imposes a group-theoretic solvability condition.