Abelian-quotient-pullbackable automorphism-invariant subgroup

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Definition

A subgroup $H$ of an abelian group $G$ is termed an abelian-quotient-pullbackable automorphism-invariant subgroup if, for every abelian-quotient-pullbackable automorphism $\sigma$ of $G$, $\sigma$ restricts to an automorphism of $H$.

Formalisms

Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

Function restriction expression	$H$ is a fully invariant subgroup of $G$ if ...	This means that full invariance is ...	Additional comments
abelian-quotient-pullbackable automorphism $\to$ function	every abelian-quotient-pullbackable automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for abelian-quotient-pullbackable automorphisms
abelian-quotient-pullbackable automorphism $\to$ endomorphism	every abelian-quotient-pullbackable automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance property for abelian-quotient-pullbackable automorphisms; i.e., it is the invariance property for abelian-quotient-pullbackable automorphism, which is a property stronger than the property of being an endomorphism
abelian-quotient-pullbackable automorphism $\to$ automorphism	every abelian-quotient-pullbackable automorphism of $G$ restricts to an automorphism of $H$	the auto-invariance property for abelian-quotient-pullbackable automorphisms; i.e., it is the invariance property for abelian-quotient-pullbackable automorphism, which is a group-closed property of automorphisms

Relation with other properties

Stronger properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Characteristic subgroup of abelian group	characteristic subgroup of abelian group			|FULL LIST, MORE INFO

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Subgroup of abelian group	subgroup of abelian group	(by definition)	subgroup of abelian group not implies abelian-quotient-pullbackable automorphism-invariant	|FULL LIST, MORE INFO

Related properties

• Abelian-quotient-pullbackable endomorphism-invariant subgroup
• Abelian-extensible automorphism-invariant subgroup
• Abelian-extensible endomorphism-invariant subgroup