TITLE: How to find subgroups of $ \;\;\Bbb Z_2\times \Bbb Z_6$
QUESTION [10 upvotes]: I am reading a first course in algebra and there is an example saying that "find all the subgroups of $\Bbb{Z}_2\times\Bbb{Z}_6$ and decide which of them are cyclic. I know that $$\Bbb{Z}_2\times\Bbb{Z}_6=\{(0,0),(0,1),\ldots,(0,5),(1,0),\ldots,(1,5)\}$$ consisting of $12$ elements and since its order is $12$, the subgroups can be of order $1$, $2$, $3$, $4$, $6$ or $12$. 
But I do not know how to find the subgroups and decide whether which of them are cyclic. Can anyone help? 
Thanks

REPLY [7 votes]: Hints:

To find the cyclic subgroups, calculate $\langle g \rangle$ for each $g \in \mathbb{Z}_2 \times \mathbb{Z}_6$.
Every group of prime order is cyclic.
Every abelian group of order $6$ is cyclic. (Alternatively, use the fact that every element of a subgroup of order $6$ must have order $1$, $2$ or $3$).
The group $\mathbb{Z}_2 \times \mathbb{Z}_6$ has exactly one subgroup of order $4$. (Use the fact that any element of the subgroup must have order $1$, $2$ or $4$).

REPLY [4 votes]: Look at some elements and see what subgroups they generate alone.
E.g. $(0,1)$ generates the $\{0\}\times\Bbb Z_6$ copy, $(0,2)$ generates a $\Bbb Z_3$ (just as $2$ in $\Bbb Z_6$), $(1,3)$ generates a $\Bbb Z_2$.
Then you can look for subgroups generated by $2$ elements, e.g. $(0,2)$ and $(1,0)$ together generate $\Bbb Z_2\times\Bbb Z_3$, and so on...