# Difference between revisions of "Rational group"

## Contents

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
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
This term is related to: linear representation theory
View other terms related to linear representation theory | View facts related to linear representation theory

## Definition

A group is termed a rational group if it satisfies the following equivalent conditions:

1. Given any two elements of the group that generate the same cyclic subgroup, the two elements are conjugate in the group. An element with the property that it is conjugate to any other element that generates the same cyclic subgroup is termed a rational element. A rational group can thus be defined as a group in which all elements are rational elements.
2. Every cyclic subgroup of the group is a fully normalized subgroup of the group.
3. If $m$ is relatively prime to the order of the group, then any element and its $m^{th}$ power are conjugate.
4. If $m$ is relatively prime to the order of an element, then the element and its $m^{th}$ power are conjugate
5. (If the group is finite): Every linear representation of the group over complex numbers has a rational-valued character.
6. (If the group is finite): Every linear representation of the group over complex numbers has an integer-valued character.
7. (If the group is finite): Every irreducible linear representation of the group over the complex numbers has a rational-valued character.
8. (If the group is finite): Every irreducible linear representation of the group over complex numbers has an integer-valued character.

## Metaproperties

### Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties

This follows from the fact that the relation of being a power as well as the relation of being conjugate can be checked coordinate-wise.

### Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties