# Finite rational group

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finite group and rational group
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## Definition

The following are some equivalent definitions:

No. Shorthand A finite group is termed a finite rational group if ... A finite group $G$ is termed a finite rational group if ...
1 rational it is a rational group. $G$ is a rational group.
2 element and powers conjugate for any $m$ relatively prime to the order of the group, every element is conjugate to its $m^{th}$ power. for $m$ relatively prime to $|G|$, and $g \in G$, the elements $g$ and $g^m$ are conjugate in $G$.
3 rational-valued character every finite-dimensional linear representation of the group over the field of complex numbers has a rational-valued character. for any finite-dimensional linear representation $\varphi:G \to GL(n,\mathbb{C})$ of $G$ over $\mathbb{C}$, the character (which we'll denote as $\chi$) of $\varphi$ satisfies the condition $\chi(g) \in \mathbb{Q}$ for all $g \in G$.
4 integer-valued character every finite-dimensional linear representation of the group over the field of complex numbers has an integer-valued character. for any finite-dimensional linear representation $\varphi:G \to GL(n,\mathbb{C})$ of $G$ over $\mathbb{C}$, the character (which we'll denote as $\chi$) of $\varphi$ satisfies the condition $\chi(g) \in \mathbb{Z}$ for all $g \in G$.
5 rational-valued character for irreducibles every irreducible linear representation of the group over the field of complex numbers has a rational-valued character. for any irreducible linear representation $\varphi:G \to GL(n,\mathbb{C})$ of $G$ over $\mathbb{C}$, the character (which we'll denote as $\chi$) of $\varphi$ satisfies the condition $\chi(g) \in \mathbb{Q}$ for all $g \in G$.
6 integer-valued character every irreducible linear representation of the group over the field of complex numbers has an integer-valued character. for any irreducible linear representation $\varphi:G \to GL(n,\mathbb{C})$ of $G$ over $\mathbb{C}$, the character (which we'll denote as $\chi$) of $\varphi$ satisfies the condition $\chi(g) \in \mathbb{Z}$ for all $g \in G$.