I thought they would be $\Bbb{Z}/n\Bbb{Z}$, where $1 \leq n \leq 10 $. What is wrong in that ? And, how is $\Bbb{Z}/n\Bbb{Z}$ a cyclic group. Is it's generator the element 1 ?
2026-04-01 13:47:52.1775051272
What are all the subgroups of $\Bbb{Z}/10\Bbb{Z}$?
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There is a bijection between subgroups of $\mathbf Z/n\mathbf Z$ and subgroups of $\mathbf Z$ that contain $n\mathbf Z$, i.e. subgroups of $\mathbf Z$ generated by divisors of $n$.
For $n=10$, you get $\mathbf Z/10\mathbf Z$, $\,2\mathbf Z/10\mathbf Z\simeq\mathbf Z/5\mathbf Z$, $\,5\mathbf Z/10\simeq \mathbf Z/2\mathbf Z\,$ and $10\mathbf Z/10\mathbf Z=0$.