# Prehomomorph-contained 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

### Definition with symbols

Suppose is a subgroup of a group . We say that is **prehomomorph-contained** in if for any surjective homomorphism of groups from a subgroup of , we have .

## Relation with other properties

### Weaker properties

- Intermediately strictly characteristic subgroup
- Strictly characteristic subgroup
- Isomorph-free subgroup
- Isomorph-containing subgroup
- Intermediately injective endomorphism-invariant subgroup
- Injective endomorphism-invariant subgroup
- Intermediately characteristic subgroup
- Characteristic subgroup
- Prehomomorph-dominated subgroup

## Metaproperties

### Intermediate subgroup condition

YES:This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition