Ambivalent group

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

Symbol-free definition

A group is said to be ambivalent if every element in it is conjugate to its inverse.

For a finite group,this is equivalent to saying that every character of the group over complex numbers, is real-valued.

An element in a group that is conjugate to its inverse is termed a real element. Thus, a group is ambivalent if and only if all its elements are real elements.

Definition with symbols

A group $G$ is said to be ambivalent if, for any $g\in G$, there exists $h\in G$ such that $hgh^{-1}=g^{-1}$.

For a finite group $G$, this is equivalent to saying that any representation $\rho :G\to GL_{n}(\mathbb {C} )$ with character $\chi$, $\chi (g)\in \mathbb {R}$ for all $g\in G$.

Examples

Extreme examples

• The trivial group is ambivalent.

Important families of groups

• Symmetric groups are ambivalent: All the symmetric groups are ambivalent.
• Classification of ambivalent alternating groups: The alternating group of degree $n$ is ambivalent only if $n\in \{1,2,5,6,10,14\}$.
• Special linear group of degree two is ambivalent iff -1 is a square
• Dihedral groups are ambivalent
• Generalized dihedral groups are ambivalent

Groups satisfying the property

Here are some basic/important groups satisfying the property:

	GAP ID
Cyclic group:Z2	2 (1)
Symmetric group:S3	6 (1)

Here are some relatively less basic/important groups satisfying the property:

	GAP ID
Alternating group:A6	360 (118)
Dihedral group:D8	8 (3)
Quaternion group	8 (4)

Here are some even more complicated/less basic groups satisfying the property:

Groups dissatisfying the property

Here are some basic/important groups that do not satisfy the property:

Here are some relatively less basic/important groups that do not satisfy the property:

	GAP ID
Alternating group:A4	12 (3)

Here are some even more complicated/less basic groups that do not satisfy the property:

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	No	ambivalence is not subgroup-closed	It is possible to have a ambivalent group $G$ and a subgroup $H$ of $G$ such that $H$ is not ambivalent.
characteristic subgroup-closed group property	No	ambivalence is not characteristic subgroup-closed	It is possible to have a ambivalent group $G$ and a characteristic subgroup $H$ of $G$ such that $H$ is not ambivalent.
quotient-closed group property	Yes	ambivalence is quotient-closed	If $G$ is an ambivalent group and $H$ is a normal subgroup of $G$, the quotient group $G/H$ is an ambivalent group.
conjugacy closed subgroup-closed group property	Yes	conjugacy-closed subgroup of ambivalent group is ambivalent	If $G$ is an ambivalent group and $H$ is a conjugacy-closed subgroup of $G$, then $H$ is ambivalent.
direct product-closed group property	Yes	ambivalence is direct product-closed	If $G_{i},i\in I$ are all ambivalent groups, so is their external direct product.
union-closed group property	Yes	ambivalence is union-closed	If a group $G$ can be expressed as a union of subgroups $H_{i},i\in I$, each of which is ambivalent, then the whole group $G$ is ambivalent.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
rational-representation group
rational group	any two elements generating the same cyclic subgroup are conjugate.	rational implies ambivalent	ambivalent not implies rational	|FULL LIST, MORE INFO
strongly ambivalent group	every non-identity element is either an involution or a product of two involutions	follows from strongly real implies real	ambivalent not implies strongly ambivalent	|FULL LIST, MORE INFO
group with two conjugacy classes	there are two conjugacy classes of elements.			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
group in which every square is a commutator	every square element is a commutator	ambivalent implies every square is a commutator	every square is a commutator not implies ambivalent	|FULL LIST, MORE INFO
group having a class-inverting automorphism	there is a class-inverting automorphism: an automorphism that sends every element to the conjugacy class of its inverse element.	For an ambivalent group, the identity automorphism is class-inverting.	class-inverting automorphism not implies ambivalent	|FULL LIST, MORE INFO
group in which every element is automorphic to its inverse	for any element of the group, there is an automorphism taking that element to its inverse.	(via group having a class-inverting automorphism)	(via group having a class-inverting automorphism)	|FULL LIST, MORE INFO
square-in-derived group	every square element is in the derived subgroup			|FULL LIST, MORE INFO

Conjunction with other properties

Property	Meaning	Result of conjunction	Proof
abelian group	any two elements commute; or equivalently, any two conjugate elements are equal.	elementary abelian 2-group	ambivalent and abelian iff elementary abelian 2-group
nilpotent group	admits a central series.	must be a nilpotent ambivalent 2-group	nilpotent and ambivalent implies 2-group
odd-order group	finite group and its order is odd.	trivial group	odd-order and ambivalent implies trivial

Facts

• Center of ambivalent group is elementary abelian 2-group
• Abelianization of ambivalent group is elementary abelian 2-group