Monomial group

A finite group is monomial if every irreducible character is a monomial character. Finite metabelian groups have this property.

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.

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.