Rational-representation group

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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.

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