# 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]
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 |

## Definition

An automorphism of a group is termed a class-preserving automorphism or class automorphism if it takes each element to within its conjugacy class. In symbols, an automorphism $\sigma$ of a group $G$ is termed a class automorphism or class-preserving 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

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Inner automorphism conjugation by an element inner implies class-preserving class-preserving not implies inner Hall-extensible automorphism, Locally inner automorphism|FULL LIST, MORE INFO
Locally inner automorphism preserves conjugacy classes of finite tuples locally inner implies class-preserving class-preserving not implies locally inner |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
automorphism that preserves conjugacy classes for a generating set there exists a generating set all of whose elements are sent to within their conjugacy class by the automorphism (obvious) preserves conjugacy classes for a generating set not implies class-preserving |FULL LIST, MORE INFO
IA-automorphism induces identity map on abelianization class-preserving implies IA IA not implies class-preserving Automorphism that preserves conjugacy classes for a generating set|FULL LIST, MORE INFO
normal automorphism sends each normal subgroup to itself class-preserving implies normal normal not implies class-preserving Extended class-preserving automorphism, Rational class-preserving automorphism|FULL LIST, MORE INFO
weakly normal automorphism sends each normal subgroup to a subgroup of itself (via normal) (via normal) Extended class-preserving automorphism, Normal automorphism, Rational class-preserving automorphism|FULL LIST, MORE INFO
extended class-preserving automorphism sends each element to conjugate or conjugate of inverse extended class-preserving not implies class-preserving |FULL LIST, MORE INFO
rational class-preserving automorphism sends each element to conjugate of element generating same cyclic group |FULL LIST, MORE INFO
center-fixing automorphism fixes every element of center class-preserving implies center-fixing center-fixing not implies class-preserving |FULL LIST, MORE INFO

## 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.