Rational group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This term is related to: linear representation theory
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Definition

No. Shorthand A group is termed a rational group if ... A group $G$ 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 $g,h \in G$ satisfy the condition that $\langle g \rangle = \langle h \rangle$, then $g$ is conjugate to $h$.
2 cyclic implies fully normalized every cyclic subgroup of the group is a fully normalized subgroup of the group. for any $g \in G$, if $C = \langle g \rangle$ is the cyclic subgroup generated by $g$, and $\sigma$ is an automorphism of $C$, there exists $v \in G$ such that conjugation by $v$ induces $\sigma$ on $C$.
3 conjugacy of appropriate powers for any element of finite order, if $m$ is relatively prime to the order of an element, then the element and its $m^{th}$ power are conjugate. For any element of infinite order, the element and its inverse must be conjugate. for any $g \in G$, if $g$ has finite order $n$, then $g$ must be conjugate to $g^m$ for all $m$ relatively prime to $n$. If $g$ has infinite order, then $g$ must be conjugate to $g^{-1}$.

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

Here is a list of examples of small order:

GAP ID
Cyclic group:Z22 (1)
Dihedral group:D88 (3)
Direct product of D8 and Z216 (11)
Elementary abelian group:E88 (5)
Finitary symmetric group of countable degree
Klein four-group4 (2)
Mathieu group:M972 (41)
Quaternion group8 (4)
Symmetric group of countable degree
Symmetric group:S36 (1)
Symmetric group:S424 (12)
Symmetric group:S6720 (763)

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 $2,3,5$, and further, the $5$-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 $G$ is a rational group and $N$ is a normal subgroup of $G$, then the quotient group $G/N$ is also a rational group.
subgroup-closed group property No rationality is not subgroup-closed It is possible to have a rational group $G$ and a subgroup $H$ of $G$ that is not rational.