Image condition

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