Finite rational group

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This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finite group and rational group
View other group property conjunctions OR view all group properties

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.