How to use a character table to get the centre

Question

I have been given a character table and I need to find from the table the centre of each character. I dont know how to do this. if someone could please explain how i can find the centre by looking at the character table.

Answer 1

Let $G$ be some finite group and $\text{irr}(G)$ the set of irreducible characters of $G$. For $\chi\in\text{irr}(G)$ define $\mathbf{Z}(\chi)=\left\{g\in G:|\chi(g)|=\chi(1)\right\}$ (called the center of the character). Then, it's a common fact that

$$\mathbf{Z}(G)=\bigcap_{\chi\in\text{irr}(G)}\mathbf{Z}(\chi)$$

So, if you are given a character table, then for each row you can look at the entries for which the modulus of that entry matches with the first entry of that row (assuming you are writing character tables with $\{1\}$ corresponding to the first column), and take the union over the conjugacy classes those entries sit below, call this union $Z_k$ if $k$ is the row we are in. Then, the above theorem says that $\mathbf{Z}(G)=Z_1\cap\cdots\cap Z_n$ if you have $n$ rows.

For proofs of the above statements you can see my blog post here, or for a more comprehensive source you can see Isaac's Character Theory of Finite Groups.

EDIT: Thankfully Yemon Choi has pointed out that you were just looking how to obtain the center of the CHARACTER from the character table, and not the center of the group. This is implicitly stated in the second paragraph of the above. Namely, to find $\mathbf{Z}(\chi)$, locate the row corresponding to $\chi$, and then take the union of the conjugacy classes lying above the row entries whose modulus equals $\chi(1)$.