# Finite rational group

From Groupprops

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 is termed a finite rational group if ... |
---|---|---|---|

1 | rational | it is a rational group. | is a rational group. |

2 | element and powers conjugate | for any relatively prime to the order of the group, every element is conjugate to its power. | for relatively prime to , and , the elements and are conjugate in . |

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 of over , the character (which we'll denote as ) of satisfies the condition for all . |

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 of over , the character (which we'll denote as ) of satisfies the condition for all . |

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 of over , the character (which we'll denote as ) of satisfies the condition for all . |

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 of over , the character (which we'll denote as ) of satisfies the condition for all . |