# Difference between revisions of "Rational group"

From Groupprops

(→Definition) |
(→Definition) |
||

Line 5: | Line 5: | ||

==Definition== | ==Definition== | ||

− | + | A [[group]] is termed a '''rational group''' if it satisfies the following equivalent conditions: | |

− | + | # Given any two elements of the group that generate the same cyclic subgroup, the two elements are [[conjugate elements|conjugate]] in the group. An element with the property that it is conjugate to any other element that generates the same cyclic subgroup is termed a [[rational element]]. A rational group can thus be defined as a group in which all elements are rational elements. | |

− | + | # Every cyclic subgroup of the group is a [[defining ingredient::fully normalized subgroup]] of the group. | |

− | An element with the property that it is conjugate to any other element that generates the same cyclic subgroup is termed a [[rational element]]. A rational group can thus be defined as a group in which all elements are rational elements. | ||

− | |||

− | |||

− | |||

− | # Every | ||

− | |||

# If <math>m</math> is relatively prime to the [[order of a group|order]] of the group, then any element and its <math>m^{th}</math> power are conjugate. | # If <math>m</math> is relatively prime to the [[order of a group|order]] of the group, then any element and its <math>m^{th}</math> power are conjugate. | ||

# If <math>m</math> is relatively prime to the [[order of an element]], then the element and its <math>m^{th}</math> power are conjugate | # If <math>m</math> is relatively prime to the [[order of an element]], then the element and its <math>m^{th}</math> power are conjugate | ||

+ | # (''If the group is finite''): Every [[linear representation]] of the group over complex numbers has a rational-valued [[defining ingredient::character of a linear representation|character]]. | ||

+ | # (''If the group is finite''): Every linear representation of the group over complex numbers has an integer-valued [[character of a linear representation|character]]. | ||

+ | # (''If the group is finite''): Every [[irreducible linear representation]] of the group over the complex numbers has a rational-valued character. | ||

+ | # (''If the group is finite''): Every irreducible linear representation of the group over complex numbers has an integer-valued [[character of a linear representation|character]]. | ||

==Examples== | ==Examples== |

## Revision as of 23:21, 19 May 2011

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This term is related to: linear representation theory

View other terms related to linear representation theory | View facts related to linear representation theory

## Definition

A group is termed a **rational group** if it satisfies the following equivalent conditions:

- Given any two elements of the group that generate the same cyclic subgroup, the two elements are conjugate in the group. An element with the property that it is conjugate to any other element that generates the same cyclic subgroup is termed a rational element. A rational group can thus be defined as a group in which all elements are rational elements.
- Every cyclic subgroup of the group is a fully normalized subgroup of the group.
- If is relatively prime to the order of the group, then any element and its power are conjugate.
- If is relatively prime to the order of an element, then the element and its power are conjugate
- (
*If the group is finite*): Every linear representation of the group over complex numbers has a rational-valued character. - (
*If the group is finite*): Every linear representation of the group over complex numbers has an integer-valued character. - (
*If the group is finite*): Every irreducible linear representation of the group over the complex numbers has a rational-valued character. - (
*If the group is finite*): Every irreducible linear representation of the group over complex numbers has an integer-valued character.

## Examples

- All symmetric groups are rational groups.
`Further information: Symmetric groups are rational` - The finitary alternating group on an infinite set is a rational group.
`Further information: Finitary alternating group on infinite set is rational` - The dihedral groups of degrees are rational groups.
`Further information: Classification of rational dihedral groups`

## Relation with other properties

### Stronger properties

- Rational-representation group: In general, the property of all linear representations being conjugate to rational representations is a stronger condition.
*For proof of the implication, refer rational-representation implies rational and for proof of its strictness (i.e. the reverse implication being false) refer rational not implies rational-representation*. - Group with two conjugacy classes

### Weaker properties

- Group in which any two elements generating the same cyclic subgroup are automorphic
- Ambivalent group
- Group in which every element is automorphic to its inverse

## Metaproperties

### Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property

View other direct product-closed group properties

This follows from the fact that the relation of being a power as well as the relation of being conjugate can be checked coordinate-wise.

### Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property

View a complete list of quotient-closed group properties

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]