<s>[INST] Characters of $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ From the Cayley table:
\begin{align*}
\begin{array}{c | c c c c }
    & (0,0) & (0,1) & (1,0) & (1,1)\\
    \hline
   (0,0) & (0,0) & (0,1) & (1,0) & (1,1)\\
   (0,1) & (0,1) & (0,0) & (1,1) & (1,0)\\
   (1,0) & (1,0) & (1,1) & (0,0) & (0,1)\\
   (1,1) & (1,1) & (1,0) & (0,1) & (0,0)\\
\end{array}
\end{align*}
How would I construct the characters of this group, $G =\mathbb{Z}_2 \oplus \mathbb{Z}_2$?
EDIT: Since $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ is abelian, all characters are one-dimensional so they take on values $\pm 1$. So we have the same character table as the Klein-4 group:
\begin{align*}
\begin{array}{c | c c c c }
 & (0,0) & (0,1) & (1,0) & (1,1)\\ 
\hline  
    \chi_{(0,0)} & 1 & 1 & 1 & 1\\
    \chi_{(0,1)} & 1 & 1 & -1 & -1\\
    \chi_{(1,0)} & 1 & -1 & 1 & -1\\
    \chi_{(1,1)} & 1 & -1 & -1 & 1\\
\end{array}
\end{align*}
If this is correct, I can put it as a solution rather than an edit however feel free to critique my attempt.
 [/INST] Since $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ is abelian, all characters are one-dimensional so they take on values $\pm 1$. So we have the same character table as the Klein-4 group:
\begin{align*}
\begin{array}{c | c c c c }
 & (0,0) & (0,1) & (1,0) & (1,1)\\ 
\hline  
    \chi_{(0,0)} & 1 & 1 & 1 & 1\\
    \chi_{(0,1)} & 1 & 1 & -1 & -1\\
    \chi_{(1,0)} & 1 & -1 & 1 & -1\\
    \chi_{(1,1)} & 1 & -1 & -1 & 1\\
\end{array}
\end{align*}
</s>