Characters of $\mathbb{Z}_2 \oplus \mathbb{Z}_2$

Question

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.

Answer 1

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*}