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 properties
VIEW 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