I am tasked in an exercise to do the following:
Construct an example of a subgroup of $\mathbb{Z_2} \times \mathbb{Z_2}$ which is not of the form $K \times J$ for some $K < \mathbb{Z_2}$ and $J < \mathbb{Z_2}$
So, it's my understanding that this question is asking us to construct a subgroup of $\mathbb{Z_2} \times \mathbb{Z_2}$ of the form $K \times J$ where either $K$ or $J$ is not a subgroup of $\mathbb{Z_2}$ correct? I know it has to have the direct product construction of $K \times J$ otherwise the elements wouldn't have the same structure, as pairs of elements from each group. This is correct? I'm a little confused at the wording, it makes it sound like we are to construct a subgroup of $\mathbb{Z_2} \times \mathbb{Z_2}$ which isn't constructed via the direct product construction.
With that out of the way, the only thought that comes to mind on potentially meeting this requirement is to have $K$ or $J$ be a subgroup of a larger group of modular arithmetic that is isomorphic to $\mathbb{Z_2}$... but if it's isomorphic to the group in question then it's just labels that distinguish it from $\mathbb{Z_2}$ and is really a distinction without a difference.
The subgroups of $\mathbb{Z_2}$ are just the trivial subgroups: $\mathbb{Z_2}$ itself and $\{[0]\}$. So I don't really see how one can get a subgroup of $\mathbb{Z_2} \times \mathbb{Z_2}$ without having each group $K,J$ for $K \times J < \mathbb{Z_2} \times \mathbb{Z_2}$, be a subgroup of $\mathbb{Z_2}$.
Any thoughts on how best to proceed?