Subgroup-conjugating automorphism
From Groupprops
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)
View other automorphism properties OR View other function properties
This is a variation of inner automorphism
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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 σ of a group is termed subgroup-conjugating if for any 
, there exists a 
such that σ(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 σ of a group G is termed a permutation-extensible automorphism if it satisfies the following:
Given any embedding of G in the symmetric group 
over a set S, there is an inner automorphism of whose restriction to G is σ.
Permutation-pushforwardable automorphism
An automorphism σ 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 over a set S, there exists an element 
such that 
, where cα is conjugation by α.
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.
