# Image condition

From Groupprops

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property

View a complete list of subgroup metaproperties

View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

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

## Definition

### Symbol-free definition

A subgroup property is said to satisfy the **image condition** if, under any surjective homomorphism, the image of a subgroup satisfying the property in the source group, again satisfies the property in the target group.

### Definition with symbols

A subgroup property is said to satisfy the image condition if whenever is a surjective homomorphism and satisfies property in , then satisfies property in .