# Rational element

From Groupprops

## Contents

## Definition

An element in a group is termed a **rational element** if, whenever is such that , there is an element such that . In other words, and are conjugate elements in .

When is a finite group, this is equivalent to the following four conditions:

- For every character of over the complex numbers, is a rational number.
- For every character of over the real numbers, is an integer.
- For relatively prime to the order of , is conjugate to .
- For relatively prime to the order of , is conjugate to .

## Relation with other properties

### Stronger properties

- Involution: An involution is an element of order two. Any element of order two is rational.

### Weaker properties

### Related group properties

- Rational group: A group in which all elements are rational.