Show that there are different subgroups of the unit circle which are isomorphic to ZxZ.
I can show there are many subgroups of the unit circle which are Isomorphic to Z but I am having no idea for this problem.
Any help will be truly appreciated.
2026-04-04 06:59:07.1775285947
Subgroups of the unit circle under complex multiplication
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For example the set $e^{p j2\pi\sqrt2+qj2\pi\sqrt3}$ under multiplication where $p$ and $q$ are integers. Since $1$, $\sqrt2$ and $\sqrt3$ are incommensurable it will be isomorphic to $Z\times Z$.
It will depend on you knowing that $1$, $\sqrt2$ and $\sqrt3$ are incommensurable (ie that $a + b\sqrt2 + c\sqrt3 = 0$ has no integer solutions). Otherwise you'll have to prove that.