# Rational element

## Definition

An element $g$ in a group $G$ is termed a rational element if, whenever $h \in G$ is such that $\langle g \rangle = \langle h \rangle$, there is an element $x \in G$ such that $xgx^{-1} = h$. In other words, $g$ and $h$ are conjugate elements in $G$.

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

1. For every character $\chi$ of $G$ over the complex numbers, $\chi(g)$ is a rational number.
2. For every character $\chi$ of $G$ over the real numbers, $\chi(g)$ is an integer.
3. For $r$ relatively prime to the order of $G$, $g$ is conjugate to $g^r$.
4. For $r$ relatively prime to the order of $g$, $g$ is conjugate to $g^r$.

## Relation with other properties

### Stronger properties

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