Monomial group: Difference between revisions

Revision as of 21:18, 10 April 2009

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:

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

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 ===Symbol-free definition===
-A [[group]] is termed '''monomial''' (or sometimes, a '''M-group''' or <math>M_1</math>-group) if every [[irreducible representation]] of the group over the complex numbers is induced from a one-dimensional representation of a subgroup. Thus, any representation can, with a suitable choice of basis, be made into a representation with all the linear transformations being expressed by monomial matrices.
+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==
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 ===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===
 * [[Solvable group]]: This follows from the [[Taketa theorem]]