Group in which every element is automorphic to its inverse

Definition
A group $$G$$ is termed a group in which every element is automorphic to its inverse if it satisfies the following equivalent conditions:


 * 1) For every element $$g \in G$$, there is an automorphism $$\sigma$$ of $$G$$ such that $$\sigma(g) = g^{-1}$$.
 * 2) There exists a group $$K$$ containing $$G$$ as a normal subgroup such that every element of $$G$$ is a defining ingredient::real element of $$K$$: it is conjugate to its inverse in $$K$$.

Stronger properties

 * Weaker than::Abelian group
 * Weaker than::Ambivalent group
 * Weaker than::Rational group
 * Weaker than::Strongly ambivalent group
 * Weaker than::Rational-representation group
 * Weaker than::Group whose automorphism group is transitive on non-identity elements
 * Weaker than::Group in which every element is order-automorphic
 * Weaker than::Group in which any two elements generating the same cyclic subgroup are automorphic

Facts

 * Alternating group implies every element is automorphic to its inverse
 * General linear group implies every element is automorphic to its inverse
 * Projective general linear group implies every element is automorphic to its inverse
 * Special linear group implies every element is automorphic to its inverse
 * Projective special linear group implies every element is automorphic to its inverse

Metaproperties
If every element of $$G$$ is automorphic to its inverse, then if $$H$$ is a characteristic subgroup of $$G$$, every element of $$H$$ is also automorphic to its inverse.