Endomorphism image

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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]

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 Retract|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