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How many subgroups does $\mathbb{Z}_{20}$ have? List a generator for each of these subgroups?

By the fundamental theorem of Cyclic group:

The subgroup of the the Cyclic group $\mathbb{Z}_{20}$ are $\left \langle a^\frac{n}{k} \right \rangle$ for all divisor k of n

The divisor k of $n=20$ are$ k=1,2,4,5,10,20$

So, the subgroups are $\left \langle a^{1} \right \rangle$,$\left \langle a^{2} \right \rangle$,$\left \langle a^{4} \right \rangle$,$\left \langle a^{5} \right \rangle$,$\left \langle a^{10} \right \rangle$,$\left \langle a^{20} \right \rangle$

Am I right? What about the generators? By the looks of it the subgroups looks like the generators itself. I'm feeling extremely confused and lost as to how I should determine the generators.

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  • $\begingroup$ By definition, $g$ is a generator of the cyclic subgroup $\langle g \rangle$. $\endgroup$ – Travis Apr 16 '16 at 14:13
  • $\begingroup$ Note, by the way, that $a^{20} = e$. $\endgroup$ – Travis Apr 16 '16 at 14:13
  • $\begingroup$ @Travis Going by the definition, $a^{k}$ is the generator for $\left \langle a^{k} \right \rangle $? $\endgroup$ – Mathematicing Apr 16 '16 at 14:14
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    $\begingroup$ It is a generator for $\langle a^k \rangle$---that's the definition of the notation $\langle \,\cdot\, \rangle$. Note that a cyclic group in general has many generators. For example, both $a^5$ and $a^{15} = a^{-5}$ are generators of $\langle a^5 \rangle$, as $(a^{15})^3 = a^5$. $\endgroup$ – Travis Apr 16 '16 at 14:16

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