Ambivalent group

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
View other properties of finite groups OR View all group properties

This term is related to: linear representation theory
View other terms related to linear representation theory | View facts related to linear representation theory

Definition

Symbol-free definition

A finite group is said to be ambivalent if it satisfies the following equivalent conditions:

• Every element is conjugate to its inverse
• Every character of the group over complex numbers, is real-valued

Definition with symbols

A finite group $G$ is said to be ambivalent if $G$ satisfies the following equivalent conditions:

• For any $g\in G$, there exists $h\in G$ such that $hg{h}^{-1}=g^{-1}$.
• For any representation $\rho :G\to G{L}_n(\mathbb{C})$ with character $\chi$, $\chi (g)\in \mathbb{R}$ for all $g\in G$.

Relation with other properties

Stronger properties

• Rational group

Weaker properties

• Group in which every square is a commutator
• Square-in-derived group

Metaproperties

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

Any quotient of an ambivalent group is ambivalent. This follows from the fact that given any element in the quotient group, we can conjugate it to its inverse by looking at the image of the conjugating element under the quotient map.

That is, if $\varphi :G\to H$ is a quotient map and $x\in H$, pick any inverse image $x^{\prime }$ of $x$ in $G$. Then, there is a $y\in G$ such that $y{x}^{\prime }{y}^{-1}=x{\text{'}}^{-1}$. Then $\varphi (y)x\varphi (y{)}^{-1}=x^{-1}$.

Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties

Any direct product of ambivalent groups is ambivalent. This follows from the fact that both the relation of being conjugate and the inverse map can be checked coordinate-wise for a direct product.

Finite unions

This group property is finite union-closed: a group which is a finite union of subgroups, each having the property, also has the property[{Category:Finite join-closed group properties]]

A group which is a union of finitely many subgroups, each of which is ambivalent, is also ambivalent. Finiteness of the union is only required so that the big group is itself finite; the condition of every element being conjugate to its inverse is preserved upon arbitrary unions.

Facts

Abelianization

Since every element of an ambivalent group is conjugate to its inverse, the image of any element in the Abelianization equals the image of its inverse. Hence, the Abelianization must be a group of exponent two, or equivalently, it must be a direct power of the cyclic group of order two.