# Inverse 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]

This article is about a general term. A list of important particular cases (instances) is available at Category: Subgroup properties satisfying inverse image condition

## Definition

### Symbol-free definition

A subgroup property is said to satisfy the **inverse image condition** if, for any homomorphism of groups, the inverse image of a subgroup satisfying the property in the group on the right, satisfies the property in the group on the left.

### Definition with symbols

Let be a subgroup property. We say that satisfies the **inverse image condition** if for any homomorphism , the following holds: whenever is a subgroup satisfying in , satisfies in .