Class-preserving automorphism

This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

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This is a variation of inner automorphism|Find other variations of inner automorphism |

Origin

Origin of the concept

The concept of class automorphism first took explicit shape when it was observed that there are automorphisms of groups that take each element to within its conjugacy class but are not inner. That is because there may not be a single element that serves uniformly as a conjugating candidate.

Origin of the term

The term class automorphism was used in the Journal of Algebra in some papers on class automorphisms.

Definition

Symbol-free definition

An automorphism of a group is termed a class automorphism if it takes each element to within its conjugacy class.

Definition with symbols

An automorphism $\sigma$ of a group $G$ is termed a class automorphism if for every $g$ in $G$, there exists an element $h$ such that $\sigma (g)=hgh^{-1}$. The choice of $h$ may depend on $g$.