# Monomial group

From Groupprops

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

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

### Symbol-free definition

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

- Every irreducible representation of the group over is induced from a one-dimensional representation of a subgroup, i.e., a linear character.
- Every finite-dimensional linear representation of the group over 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