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 $g\in G$.