What Are the Subgroups of Z3 x Z3?

[FanofAFan]

Homework Statement

Find all of the subgroups of Z₃ x Z₃

Homework Equations

Z₃ x Z₃ is isomorphic to Z₉

The Attempt at a Solution

x = (0,1,2,3,4,5,6,7,8)
<x⁰> or just <0> = {0}
<1> = {identity}
<2> = {0,2,4,6} also wasn't sure if I did this one correctly x o x for x²
<3> = {0,3,6}
and so on until I got
<8> = {0,8}
<9> = {0}

I feel like I might be completely wrong but is this even a cyclic group, and do I need to approach finding the subgroups differently? Thanks

 

[Dick]

Z3xZ3 is not cyclic. Any element of Z3xZ3 has order 3. And even if it were, your subgroups of Z9 have problems.

 

[Office_Shredder]

If you're going to compute the subgroup of Z₉ generated by 2, you can't stop at 8. We start with 0, 2, 4, 6, 8, and then 2+8=10=1 mod 9, 1+2=3, then you get 5 and 7 so 2 generates the whole group.

Of course like Dick said those aren't the subgroups you're looking for anyway