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 |

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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 $\sigma$ of a group is termed subgroup-conjugating if for any $H\leq G$, 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 $\alpha$.

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

• Normal 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

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