Class-inverting automorphism implies every element is automorphic to its inverse

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group having a class-inverting automorphism) must also satisfy the second group property (i.e., group in which every element is automorphic to its inverse)
View all group property implications | View all group property non-implications
Get more facts about group having a class-inverting automorphism|Get more facts about group in which every element is automorphic to its inverse

Statement

Suppose $G$ is a group having a Class-inverting automorphism (?), i.e., there exists an automorphism $\sigma$ of $G$ such that for every $g\in G$, there exists $x\in G$ such that $\sigma (g)=x{g}^{-1}{x}^{-1}$. Then, $G$ is a Group in which every element is automorphic to its inverse (?): for every $g\in G$, there exists an automorphism $\alpha$ of $G$ such that $\alpha (g)=g^{-1}$.

Related facts

• General linear group implies every element is automorphic to its inverse
• Special linear group implies every element is automorphic to its inverse
• Alternating group implies every element is automorphic to its inverse
• Every element is automorphic to its inverse is characteristic subgroup-closed
• Normal subgroup of ambivalent group implies every element is automorphic to its inverse