# 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.

• Since $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ is isomorphic to the Klein-4 group, does this imply the character table will be the same? – Math Feb 13 at 12:06
• Yes isomorphic groups have isomorphic character tables. Only possibly the rows and columns would be permuted if you had put the group elements and irreducible characters in a different order. – Ben Feb 13 at 12:25
• @Ben just to check, is my orderings correct? – Math Feb 13 at 12:35
• I don’t know your definition of the Klein four group, but the character table you wrote seems to just be the character table for $Z/2\oplus Z/2$, so you don’t have to worry about an isomorphism. – Ben Feb 13 at 12:43

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*}