Rational-representation group

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Rational-representation group, all facts related to Rational-representation group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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

A rational-representation group is a finite group satisfying the following properties:

1. The field of rational numbers $\mathbb{Q}$ is a splitting field, i.e., every irreducible representation in characteristic zero is realizable over the rational numbers.
2. Every irreducible representation in characteristic zero can be realized over $\mathbb{Z}$.
3. All its characters are rational-valued.
4. All its characters are integer-valued.

Equivalence of definitions

Definitions (1) and (2) are equivalent because linear representation is realizable over principal ideal domain iff it is realizable over field of fractions.

Relation with other properties

Weaker properties

• Rational group: A rational group is a finite group such that all its characters are rational-valued (hence integer-valued). The quaternion group is an example of a rational group that is not a rational-representation group.
• Ambivalent group: A finite group in which every character is real-valued.

Facts

• Symmetric groups on finite sets are rational-representation groups. For full proof, refer: Symmetric groups are rational-representation groups
• Dihedral group:D8 is a rational-representation group.
• Trivial group, cyclic group:Z2, and dihedral group:D8 are the only three groups with the property that all their irreducible representations can be written over the rationals as orthogonal matrices. For full proof, refer: Classification of groups for which every irreducible representation can be realized using orthogonal matrices over the rational numbers