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.

Weaker properties

• Real element

Related group properties

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