Abelian-extensible endomorphism-invariant subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
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Definition

A subgroup $H$ of an abelian group $G$ is termed an abelian-extensible endomorphism-invariant subgroup if it is invariant under all the abelian-extensible endomorphisms of $G$ .

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

Relation with other properties

Stronger properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Fully invariant subgroup of abelian group
Subgroup of finite abelian group

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Subgroup of abelian group

Related properties