# Subgroup-conjugating 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 is a variation of inner automorphism|Find other variations of inner automorphism |

## Contents |

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## Definition

### Symbol-free definition

An automorphism of a group is termed **subgroup-conjugating** if, under the action of this automorphism, each subgroup goes to a conjugate subgroup.

### Definition with symbols

An automorphism of a group is termed **subgroup-conjugating** if for any , there exists a such that .

## Alternative definitions

The following notions, **permutation-extensible automorphism**, and **permutation-pushforwardable automorphism**, turn out to be equivalent to subgroup-conjugating automorphism. `For full proof, refer: Equivalence of definitions of subgroup-conjugating automorphism`

### Permutation-extensible automorphism

An automorphism of a group is termed a permutation-extensible automorphism if it satisfies the following:

Given any embedding of in the symmetric group over a set , there is an inner automorphism of whose restriction to is .

### Permutation-pushforwardable automorphism

An automorphism of a group is termed a permutation-extensible automorphism if it satisfies the following:

Given any homomorphism from to the symmetric group over a set , there exists an element such that , where is conjugation by .

## Relation with other properties

### Stronger properties

- Inner automorphism:
`For full proof, refer: Inner implies subgroup-conjugating` - Extensible automorphism:
`For full proof, refer: Extensible implies subgroup-conjugating` - Strong power automorphism

### Weaker properties

## 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

The subgroup-conjugating automorphisms of a group form a subgroup of its automorphism group.