Monomial group (reviewed)

A group $G$ is monomial or an M-group if every complex irreducible representation of $G$ is induced from a $1$-dimensional representation of some subgroup. Such a group is always solvable, and monomial groups include all supersolvable groups and solvable A-groups. A purely group-theoretic description of monomial groups is not known.

While every supersolvable group is monomial, the converse does not hold, as shown by these examples. Similarly, these examples show that not every solvable A-group is supersolvable. There are also examples of solvable groups that are not monomial.