Isomorphic subgroups to $Z$

Question

Find all subgroups of $Z \times Z_2 $ which are isomorphic to Z.

I think the subgroups which are isomorphic to Z are $nZ \times \{0\}$ and $nZ \times \{1\}$ but I am not sure if there are more.

Thank you

Answer 1

The subsets $n\Bbb Z\times \{1\}$ are not subgroups. Any element of $G=\Bbb Z\times\Bbb Z_2$ other than $(0,0)$ or $(0,1)$ generates a subgroup isomorphic to $\Bbb Z$. Then $(n,0)$ generates $n\Bbb Z\times\{ \overline0\}$ for $n\ne0$, while $(n,\overline1)$ generates $\{(kn,\overline k):k\in\Bbb Z\}$. Here I'm using $\overline k$ to denote the "reduction modulo $2$" of $k$ in $\Bbb Z_2$.