# 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]

## 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
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