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Contents 1 Origin 1.1 Origin of the concept 1.2 Origin of the term 2 Definition 2.1 Symbol-free definition 2.2 Definition with symbols 3 Relation with other properties 3.1 Stronger properties 3.2 Weaker properties 3.3 Related properties 4 Metaproperties 4.1 Group-closedness 4.2 Direct product-closedness

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
View other semistandard definitions in group theory

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)
View other automorphism properties OR View other function properties

This is a variation of inner automorphism
View a complete list of variations of inner automorphism OR read a survey article on varying 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 σ of a group G is termed a class automorphism if for every g in G, there exists an element h such that σ(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 G₁ and G₂ be groups and σ₁,σ₂ be class automorphisms of G₁,G₂ respectively. Then, is a class automorphism of .

Here, is the automorphism of that acts as σ₁ on the first coordinate and σ₂ on the second.