Monomial group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

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 monomial linear representation: it is a direct sum of representations induced from one-dimensional representations of subgroups.

Relation with other properties

Stronger properties

• Elementary group
• Finite nilpotent group
• Finite supersolvable group: For full proof, refer: Finite supersolvable implies monomial

Weaker properties

• Solvable group: This follows from the Taketa theorem