Inverse 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

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Inverse image condition, all facts related to Inverse image condition) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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 $p$ be a subgroup property. We say that $p$ satisfies the inverse image condition if for any homomorphism $\phi: G \to H$ , the following holds: whenever $N$ is a subgroup satisfying $p$ in $H$ , $\phi^{-1}(N)$ satisfies $p$ in $G$ .