Find the number of subgroups of $\mathbb{Z}/(5)\times \mathbb{Z}/(5)$ including trivial subgroups.
My Work:
If we consider $\mathbb{Z}/(5)$ then the only subgroups are trivial subgroups. But how can we find it for $\mathbb{Z}/(5)\times \mathbb{Z}/(5)$? Is their a particular Proposition, Theorem or corollary which can be used here? Please help.
The group has the identity plus $24$ elements of order $5$. When you have a subgroup of order $5$, you have a collection of the identity, plus four of the other $24$ elements. So do you see how the $24$ elements are partitioned?