Class-preserving automorphism

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 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 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 $g$.

Relation with other properties

Stronger properties

• Inner automorphism: For proof of the implication, refer Inner implies class and for proof of its strictness (i.e. the reverse implication being false) refer Class not implies inner.

Weaker properties

• IA-automorphism
• Center-fixing automorphism

Related properties

• Subgroup-conjugating automorphism

Metaproperties

Group-closedness

This automorphism property is group-closed: it is closed under the group operations on automorphisms (composition, inversion and the identity map). It follows that the subgroup comprising automorphisms with this property, is a normal subgroup of the automorphism group
View a complete list of group-closed automorphism properties

Clearly, a product of class automorphisms is a class automorphism, and the inverse of a class automorphism is a class automorphism. Thus, the class automorphisms form a group which sits as a subgroup of the automorphism group. Moreover, this subgroup contains the group of inner automorphisms, and is a normal subgroup inside the automorphism group.

Direct product-closedness

This automorphism property is direct product-closed
View a complete list of direct product-closed automorphism properties

Let $G_{1}$ and $G_{2}$ be groups and $\sigma _{1},\sigma _{2}$ be class automorphisms of $G_{1},G_{2}$ respectively. Then, $\sigma _{1}\times \sigma _{2}$ is a class automorphism of $G_{1}\times G_{2}$.

Here, $\sigma _{1}\times \sigma _{2}$ is the automorphism of $G_{1}\times G_{2}$ that acts as $\sigma _{1}$ on the first coordinate and $\sigma _{2}$ on the second.