Endomorphism image

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 a group $G$ is termed an endomorphism image if there exists an endomorphism of $G$ whose image is precisely the subgroup $H$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
direct factor				|FULL LIST, MORE INFO
retract	image of an idempotent endomorphism			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
divisibility-closed subgroup	if every element of the group has a $n^{th}$ root, so does every element of the subgroup.	endomorphism image implies divisibility-closed	any subgroup of a finite group that is not an endomorphism image works.	|FULL LIST, MORE INFO
powering-invariant subgroup	if every element of the group has a unique $n^{th}$ root, so does every element of the subgroup.	(via divisibility-closed)	(via divisibility-closed)	|FULL LIST, MORE INFO

Related properties

• Endomorphism kernel: Kernel of an endomorphism.