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)
View other automorphism properties OR View other function properties
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 of a group is termed subgroup-conjugating if for any , there exists a such that .
Relation with other properties
Stronger properties
- Inner automorphism: For full proof, refer: Inner implies subgroup-conjugating
- Permutation-extensible automorphism: For full proof, refer: Permutation-extensible implies subgroup-conjugating
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.