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Pushforwardable 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 term is related to: Extensible Automorphisms Problem
See more terms related to Extensible Automorphisms Problem OR see facts/theorems related to Extensible Automorphisms Problem
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This is a variation of inner automorphism
View a complete list of variations of inner automorphism OR read a survey article on varying inner automorphism
Definition
Symbol-free definition
An automorphism of a group is termed pushforwardable if it can be pushed forward via any homomorphism from that group to any group.
Definition with symbols
An automorphism σ of a group G is termed extensible if, for any homomorphism π from G to any group H, there is an automorphism φ of H such that π.σ = φ.π.
In terms of the pushforwardability operator
The automorphism property of being pushforwardable is the result of applying the pushforwardability operator to the tautology.
Relation with other properties
This property is conjectured to equal the property: inner automorphism
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 pushforwardable automorphisms of a group form a subgroup of its automorphism group. This follows from the fact that the pushforward of the product of two automorphisms can be taken as the product of their pushforwards, and the pushforward of the inverse of an automorphism can be taken as the inverse of the pushforward.
