Monomial group

Symbol-free definition
A finite group is termed monomial (or sometimes, a M-group or $$M_1$$-group) with respect to a field $$k$$ (whose characteristic does not divide the group order) if it satisfies the following equivalent conditions:


 * 1) Every irreducible representation of the group over $$k$$ is induced from a one-dimensional representation of a subgroup, i.e., a linear character.
 * 2) Every finite-dimensional linear representation of the group over $$k$$ is a defining ingredient::monomial linear representation: it is a direct sum of representations induced from one-dimensional representations of subgroups.

Stronger properties

 * Weaker than::Elementary group
 * Weaker than::Finite nilpotent group
 * Weaker than::Finite supersolvable group:

Weaker properties

 * Solvable group: This follows from the Taketa theorem