# Rational-representation group

## Contents

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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
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## Definition

A rational-representation group is a finite group satisfying the following equivalent 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 $\mathbb{Z}$.

### 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 $\{ 1 \}$
cyclic group:Z2 2 1,1 $\{ 1, -1 \}$
Klein four-group 4 1,1,1,1, $\{ 1,-1 \}$
symmetric group:S3 6 1,1,2 $\{ 1,0,-1 \}$
elementary abelian group:E8 8 1,1,1,1,1,1,1,1 $\{ 1, -1 \}$
dihedral group:D8 8 1,1,1,1,2 $\{ 1,0,-1 \}$
direct product of S3 and Z2 12 1,1,1,1,2,2 $\{ 1,0,-1 \}$
elementary abelian group:E16 16 1 (16 times) $\{ 1,-1 \}$
direct product of D8 and Z2 16 1,1,1,1,1,1,1,1,2,2 $\{ 1,0,-1 \}$
symmetric group:S4 24 1,1,2,3,3 $\{ 1,0,-1 \}$
direct product of S3 and V4 24 1 (8 times), 2 (4 times) $\{ 1,0,-1 \}$

## 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.