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Is there any non trivial example of subgroup of a cyclic group?

Every subgroup of a cyclic group is cyclic. I couldnt find any non trivial examples of it since every subgroup would have the generator and if the generator is there then the entire group is there.

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$\{0 + 4\mathbb Z, 2 + 4\mathbb Z\} \subseteq Z/4\mathbb Z$ for a finite example, or think of $n\mathbb Z \subseteq \mathbb Z$. – martini Dec 10 '13 at 12:43
Wow i totally missed that thanks. – vinothkr Dec 10 '13 at 12:44
And from the answer below, you conclude that if the order is prime, there is no proper subgroup at all. – Jean-Claude Arbaut Dec 10 '13 at 12:46
That was going to be my next question. You read my mind :) – vinothkr Dec 10 '13 at 12:47
up vote 5 down vote accepted

Not every subgroup has to contain the generator. Consider the cyclic group of order $4$, which we denote as $\{1,a,a^2,a^3\}$. Then $\{1,a^2\}$ is a proper subgroup.

More generally, if $\langle a \rangle$ is a cyclic group of order $x$ and $d$ divides $x$, we have a cyclic subgroup $\langle a^d \rangle$.

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+1 for the general. Nice – Eleven-Eleven Dec 10 '13 at 13:13

If your group is $G=(\mathbb{Z},+)$, your generator is $1$. If your subgroup is $H=(2\mathbb{Z},+)$, the generator is $2$ which is not the generator for $G$.

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