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 defining ingredient::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 $$g^r$$.

Stronger properties

 * Weaker than::Involution: An involution is an element of order two. Any element of order two is rational.

Weaker properties

 * Stronger than::Real element

Related group properties

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