This question is from a past exam.
Find all distinct subgroups of $\mathbb{Z}_4 \times \mathbb{Z}_4$ isomorphic to $\mathbb{Z}_4$
Attempt/Thoughts?
Since $\mathbb{Z}_4$ is cyclic we are looking for cyclic subgroups of the given group. Then can I use the fundamental theorem of f.g.ab. groups to solve this problem?. I have a hard time imagining how the elements in $\mathbb{Z}_4\times \mathbb{Z}_4$ look like. Is there a standard way to proceed in this type of problem?.
Can somebody help?. I usually don't ask for detailed answers in here. But in this case I would really appreciate it. Thanks.