Ambivalent group

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 defining ingredient::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 $$g \in G$$.

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

Facts

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