# Image-potentially fully invariant subgroup

From Groupprops

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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

Suppose is a subgroup of a group . We say that is an **image-potentially fully invariant subgroup** of if there exists a group , a surjective homomorphism , and a subgroup of such that .

## 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 |