Subgroup-conjugating automorphism

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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.

Relation with other properties

Stronger properties

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.