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

• 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 $1,2,3,4,6$ are rational groups. Further information: Classification of rational dihedral groups

Here is a list of examples of small order:

	GAP ID
Cyclic group:Z2	2 (1)
Dihedral group:D8	8 (3)
Direct product of D8 and Z2	16 (11)
Elementary abelian group:E8	8 (5)
Finitary symmetric group of countable degree
Klein four-group	4 (2)
Mathieu group:M9	72 (41)
Quaternion group	8 (4)
Symmetric group of countable degree
Symmetric group:S3	6 (1)
Symmetric group:S4	24 (12)
Symmetric group:S6	720 (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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
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.