The problem is:

Construct the group table for the additive group $\mathbb{Z\_3}$, and show that $\mathbb{Z\_3}$ is a subgroup of $\mathbb{Z\_9}$. \begin{array}{|c|c|c|c|} \hline
+& \overline0 & \overline1 & \overline2 \\ \hline \overline0 & & &\\ \hline \overline1 & & &\\ \hline \overline2 & & &\\ \hline \end{array}

I was able to construct the table as 
\begin{array}{|c|c|c|c|} \hline
+& \overline0 & \overline1 & \overline2 \\ \hline \overline0 &0 &1 &2\\ \hline \overline1 &1 &2 &0\\ \hline \overline2 &2 &0 &1\\ \hline \end{array}
However, I'm having troubles showing that that $\mathbb{Z\_3}$ is a subgroup of $\mathbb{Z\_9}$. How can this be done?

Also, I'm not sure if I should overline the numbers in the table as well or I can keep them like this.