# Image condition

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 metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

## Contents

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 $p$ is said to satisfy the image condition if whenever $\phi:G \to K$ is a surjective homomorphism and $H \le G$ satisfies property $p$ in $G$, then $\phi(H)$ satisfies property $p$ in $K$.