Monomial group: Difference between revisions
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{{group property}} | {{group property}} | ||
{{basicdef in|linear representation theory}} | |||
==Definition== | ==Definition== | ||
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[group]] is termed '''monomial''' (or sometimes, a '''M-group''') if | A [[finite group]] is termed '''monomial''' (or sometimes, a '''M-group''' or <math>M_1</math>-group) with respect to a field <math>k</math> (whose characteristic does not divide the group order) if it satisfies the following equivalent conditions: | ||
# Every [[irreducible representation]] of the group over <math>k</math> is induced from a one-dimensional representation of a subgroup, i.e., a [[linear character]]. | |||
# Every finite-dimensional linear representation of the group over <math>k</math> is a [[defining ingredient::monomial linear representation]]: it is a direct sum of representations induced from one-dimensional representations of subgroups. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Finite nilpotent group]] | * [[Weaker than::Elementary group]] | ||
* [[Finite supersolvable group]] | * [[Weaker than::Finite nilpotent group]] | ||
* [[Weaker than::Finite supersolvable group]]: {{proofat|[[Finite supersolvable implies monomial]]}} | |||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Solvable group]]: This follows from the [[Taketa theorem]] | * [[Solvable group]]: This follows from the [[Taketa theorem]] | ||
Latest revision as of 22:48, 10 November 2023
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
This article gives a basic definition in the following area: linear representation theory
View other basic definitions in linear representation theory |View terms related to linear representation theory |View facts related to linear representation theory
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