# Image-potentially fully invariant subgroup

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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

Suppose $H$ is a subgroup of a group $G$. We say that $H$ is an image-potentially fully invariant subgroup of $G$ if there exists a group $K$, a surjective homomorphism $\rho:K \to G$, and a subgroup $L$ of $K$ such that $\rho(L) = H$.

## Relation with other properties

### Stronger properties

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

### Weaker properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Normal subgroup