Image condition

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 $$K$$.