Find all sub groups of order 4 in $\mathbf{Z}_4 \oplus \mathbf{Z}_4$.
Solution : $\mathbf{Z}_4 =\{0,1,2,3\}$
$O(1) = O(3) = 4$, $O(0) = 1$, $O(2) = 2$
Hence, I found the subgroups of order 4 as follows:
$\langle 1,0 \rangle,\langle 0,1 \rangle,\langle 0,\mathbf{Z}_4 \rangle,\langle \mathbf{Z}_4, 0 \rangle,\langle 1,1 \rangle,\langle 1,3 \rangle,\langle 3,1 \rangle,\langle 3,1 \rangle,\langle 1,2 \rangle,\langle 2,1 \rangle$.
So, all of these are cyclic sub groups. How do we know that there are only cyclic subgroups in here and not some non-cyclic groups as well?
$\mathbf{Z}_4 \oplus \mathbf{Z}_4$ is not cyclic since $\gcd(|\mathbf{Z}_4|,|\mathbf{Z}_4|) \neq 1$, hence there can be non-cyclic subgroups of order 4 as well. How do we make sure apart from directly computing?
Hint :