Rational group
From Groupprops
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 properties
VIEW 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
No. | Shorthand | A group is termed a rational group if ... | A group is termed a rational group if ... |
---|---|---|---|
1 | conjugacy of elements generating same cyclic subgroup | 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. | if satisfy the condition that , then is conjugate to . |
2 | cyclic implies fully normalized | every cyclic subgroup of the group is a fully normalized subgroup of the group. | for any , if is the cyclic subgroup generated by , and is an automorphism of , there exists such that conjugation by induces on . |
3 | conjugacy of appropriate powers | for any element of finite order, if is relatively prime to the order of an element, then the element and its power are conjugate. For any element of infinite order, the element and its inverse must be conjugate. | for any , if has finite order , then must be conjugate to for all relatively prime to . If has infinite order, then must be conjugate to . |
There are a number of other equivalent definitions that can be used in the case that we are dealing with a finite rational group.
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
Here is a list of examples of small order:
Relation with other properties
Conjunction with other properties
Other property | Result of conjunction | Proof/explanation |
---|---|---|
abelian group | elementary abelian 2-group, i.e., a vector space over the field of two elements | rational and abelian implies elementary abelian 2-group |
nilpotent group | must be a rational 2-group | rational and nilpotent implies 2-group (see also odd-order and ambivalent implies trivial) |
finite solvable group | only prime factors that can divide the order are , and further, the -Sylow subgroup must be a normal Sylow subgroup and also must be elementary abelian. |
Hegedűs Pál proved ([1]) in 2004, the following results: "In this paper it is proved that in a solvable rational group the Sylow 5-subgroup is always normal and elementary Abelian. Moreover, the structure of rational {2, 5}-groups is described in detail."
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
rational-representation group (also called strongly rational group) | the field of rational numbers is a splitting field | rational-representation implies rational | rational not implies rational-representation | |FULL LIST, MORE INFO |
group with two conjugacy classes |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
group in which any two elements generating the same cyclic subgroup are automorphic | if two elements generate the same cyclic subgroup, there is an automorphism of the whole group sending one to the other. | (follows from the fact that conjugations are inner automorphisms) | all abelian groups that are not elementary abelian 2-groups give examples | |
ambivalent group | every element is conjugate to its inverse | rational implies ambivalent | ambivalent not implies rational | |FULL LIST, MORE INFO |
group in which every element is automorphic to its inverse | Ambivalent group, Group in which any two elements generating the same cyclic subgroup are automorphic|FULL LIST, MORE INFO |
Metaproperties
Metaproperty | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
direct product-closed group property | Yes | rationality is direct product-closed | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] |
quotient-closed group property | Yes | rationality is quotient-closed | If is a rational group and is a normal subgroup of , then the quotient group is also a rational group. |
subgroup-closed group property | No | rationality is not subgroup-closed | It is possible to have a rational group and a subgroup of that is not rational. |