Finding subgroups of Factor/ Quotient Groups

[porroadventum]

Homework Statement

Describe all the subgroups of Z/9Z. How many are there? Describe all the subgroups of Z/3ZxZ/3Z. How many are there?

The Attempt at a Solution

I don't even know where to start with this question. If someone could just point me in the right direction that would be great.

Thank you.

 

[porroadventum]

I have made an attempt, if someone could let me know if it is correct or not, that would be much appreciated!

The elements of Z/9Z are {0,1,2,3,4,5,6,7,8} with operation modulo9.

The elements (1,2,4,5,7,8} have order 9 and generate the whole group.
{0,3,6} has order 3.

Therefore there are two subgroups.

 

[jbunniii]

I have made an attempt, if someone could let me know if it is correct or not, that would be much appreciated!

The elements of Z/9Z are {0,1,2,3,4,5,6,7,8} with operation modulo9.

The elements (1,2,4,5,7,8} have order 9 and generate the whole group.
{0,3,6} has order 3.

Therefore there are two subgroups.

{0} is also a subgroup.

 

[Joffan]

It looks like you're good for $Z/9Z$.

$Z/3Z \times Z/3Z$ is {(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)} - which is not cyclic.