# Rational-representation group

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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 propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

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

- The field of rational numbers is a splitting field, i.e., every irreducible representation in characteristic zero is realizable over the rational numbers.
- Every irreducible representation in characteristic zero can be realized over .

### 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.

## Examples

Here are some small examples, with relevant contextual information.

Group | Order | List of degrees of irreducible representations | List of entries of matrices arising across all irreducible representations when chosen with simplest entries |
---|---|---|---|

trivial group | 1 | 1 | |

cyclic group:Z2 | 2 | 1,1 | |

Klein four-group | 4 | 1,1,1,1, | |

symmetric group:S3 | 6 | 1,1,2 | |

elementary abelian group:E8 | 8 | 1,1,1,1,1,1,1,1 | |

dihedral group:D8 | 8 | 1,1,1,1,2 | |

direct product of S3 and Z2 | 12 | 1,1,1,1,2,2 | |

elementary abelian group:E16 | 16 | 1 (16 times) | |

direct product of D8 and Z2 | 16 | 1,1,1,1,1,1,1,1,2,2 | |

symmetric group:S4 | 24 | 1,1,2,3,3 | |

direct product of S3 and V4 | 24 | 1 (8 times), 2 (4 times) |

## Facts

- Symmetric groups on finite sets are rational-representation groups.
`For full proof, refer: Symmetric groups are rational-representation` - 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 finite groups for which every irreducible representation can be realized using orthogonal matrices over the rational numbers`

## 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.