Projectivity

Definition
Suppose $$G$$ and $$H$$ are groups. Suppose $$f:G \to H$$ is a bijective set map such that for any subset $$A$$ of $$G$$, $$A$$ is a subgroup of $$G$$ if and only if $$f(A)$$ is a subgroup of $$H$$. Then, $$f$$ induces a map $$\overline{f}$$ from the lattice of subgroups of $$G$$ to the lattice of subgroups of $$H$$. A map between lattices of subgroups obtained in this way is termed a projectivity.