List the elements of the cyclic subgroup $<(13, 3)>$ in $\mathbb Z_{26}$ x $\mathbb Z_{9}$

Question

List the elements of the cyclic subgroup $<(13, 3)>$ in $\mathbb Z_{26}$ x $\mathbb Z_{9}$

I'm not entirely sure I am doing this problem right. Do I simply add (13,3) until I reach (0,0)?

I did this:

$<(13,3)>={(0,0), (13,3), (0,3), (13,6)}$

Is this correct? Thanks

Answer 1

Yes, you keep on adding $(13,3)$ until you reach $(0,0)$.

But do it carefully...

$$ (0,0) \to (13,3) \to (0,6) \to (13,0) \to (0,3) \to (13,6) \to (0,0) $$

Since $13$ has order $2$ mod $26$ and $3$ has order $3$ mod $9$, you expect $(13,3)$ to have order $6=lcm(2,3)$, which is what you see.

You could also do it separately, which probably will help avoiding getting confused about which modulus to use:

$\qquad 0 \to 13 \to 0 \to 13 \to 0 \to 13 \to 0 \quad \pmod {26} $

$\qquad 0 \to \hphantom1 3 \to 6 \to \hphantom1 0 \to 3 \to \hphantom1 6 \to 0 \quad \pmod 9 $