# 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 |

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## 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 $\sigma$ of a group is termed subgroup-conjugating if for any $H \le G$, there exists a $g \in G$ such that $\sigma(H) = gHg^{-1}$.

## 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 $\sigma$ of a group $G$ is termed a permutation-extensible automorphism if it satisfies the following:

Given any embedding of $G$ in the symmetric group $\operatorname{Sym}(S)$ over a set $S$, there is an inner automorphism of $\operatorname{Sym}(S)$ whose restriction to $G$ is $\sigma$.

### Permutation-pushforwardable automorphism

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

Given any homomorphism $f$ from $G$ to the symmetric group $\operatorname{Sym}(S)$ over a set $S$, there exists an element $\alpha \in \operatorname{Sym}(S)$ such that $c_\alpha \circ f = f \circ \sigma$, where $c_\alpha$ is conjugation by $\alpha$.

## 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.