# Subgroups of $\mathbb{Z} \times \mathbb{Z}_2$

Is it true that the only non-trivial (subgroups other than the trivial group and the group itself) subgroups of $\mathbb{Z} \times \mathbb{Z}_2$ are all isomorphic to $\mathbb{Z}$? My intuition tells me this is true, but is there any formal way to see this?

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$n\mathbb{Z}\times\mathbb{Z}_2$ –  yoyo Mar 21 at 22:43
More in particular , the non-trivial subgroup $\,\{0\}\times \Bbb Z_2\cong \Bbb Z_2\ncong \Bbb Z\,$ ... –  DonAntonio Mar 21 at 22:50
I assume $\mathbb{Z}_2$ is again the wrong notation for $\mathbb{Z}/2\mathbb{Z}$? –  Martin Brandenburg Mar 22 at 20:50

Well you have $\Bbb{Z}_{2}$, of course, and more generally all subgroups of the form $n \Bbb{Z} \times \Bbb{Z}_{2}$, which are definitely not isomorphic to $\Bbb{Z}$.

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There is an well known fact proposed by Kaplansky that says:

If $G$ is an infinite group which is isomorphic to every proper subgroup, then $G\cong \mathbb Z$.

If your group has this property so it should be $\mathbb Z$, but as you see via @Andreas's answer, it is wrong.

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+1 for adding an important theorem to know! +1 just because! ;-) –  amWhy Mar 23 at 0:42