Hello Mathematics Community. I am unsure about how to solve this problem involving the number of subgroups in an abelian group.
How many subgroups of order $p^2$ does the abelian group $\mathbb{Z}_p \oplus \mathbb{Z}_p$ have?
I have first tried to use the First Isomorphism Theorem, but I do not think it helped. Then I considered the following:
The order of our group is $|\mathbb{Z}_p \oplus \mathbb{Z}_p|=p^2$. By the structure theorem of finite abelian groups, then $\mathbb{Z}_p \oplus \mathbb{Z}_p \cong \mathbb{Z}_{p^2}$. Now I am stuck and do not know how to proceed, or whether this is the right direction to solving the problem.
As always, any help is greatly appreciated; thanks in advance.
The big group has, as you pointed out, only $p^2$ elements!