I read on A course in arithmetics, of J.P. Serre, that the number of nontrivial subgroups of $(\mathbb Z/p\mathbb Z)^2$, with $p$ prime, is equal to "the number of points on the projective line over a field with $p$ elements". This is stated without proof, and I cannot understand it.
The projective line it mentions is the quotient $$(\mathbb Z/p\mathbb Z\setminus\{0\})^2/\sim, \quad \text{where }a\sim b\Leftrightarrow \lambda a=b,\ \text{ for a }\lambda \in \mathbb Z/p\mathbb Z\setminus \{ 0 \}.$$ What is the relation between this quotient and the subgroups of $(\mathbb Z/p\mathbb Z)^2$?
The (additive) subgroups of $\Bbb F_p^2$ are exactly the $\Bbb Z$-submodules of $\Bbb F_p^2$. Moreover, the action $(n, v)\mapsto nv$ of $\Bbb Z$ on $\Bbb F_p^2$ coincides, up to taking the quotient by $p\Bbb Z$ on $\Bbb Z$, with the vector space action $(n+p\Bbb Z, v)\mapsto nv$ of $\Bbb F_p$ on $\Bbb F_p^2$. Therefore, the non-zero, cyclic subgroups of of $\Bbb F_p^2$ are exactly the one-dimensional vector subspaces of $\Bbb F_p^2$. Or, equivalently, the points of $\Bbb P^1(\Bbb F_p)$.