# 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

### Groups satisfying the property

Here are some basic/important groups satisfying the property:

GAP ID
Cyclic group:Z22 (1)
Klein four-group4 (2)
Symmetric group:S36 (1)

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

GAP ID
Alternating group:A560 (5)
Alternating group:A6360 (118)
Dihedral group:D88 (3)
Quaternion group8 (4)
Symmetric group:S424 (12)
Symmetric group:S6720 (763)

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

GAP ID
Dihedral group:D1616 (7)
Direct product of D8 and Z216 (11)
Mathieu group:M972 (41)

### 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:A412 (3)

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

GAP ID
Alternating group:A7
M1616 (6)
Semidihedral group:SD1616 (8)

## 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. Rational group|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) Group having a class-inverting automorphism|FULL LIST, MORE INFO
square-in-derived group every square element is in the derived subgroup Group in which every square is a commutator|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