Monomial group: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[group]] is termed '''monomial''' (or sometimes, a '''M-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 [[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. | ||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 01:07, 31 December 2007
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
Definition
Symbol-free definition
A group is termed monomial (or sometimes, a M-group or -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.
Relation with other properties
Stronger properties
Weaker properties
- Solvable group: This follows from the Taketa theorem