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.

References

• Subgroup Lattices of Groups (de Gruyer Expositions in Mathematics) by Roland Schmidt, Page 17, ^{More info}