# How to use a character table to get the kernel

I have been given a character table and I need to find from the table the kernel of each character. I dont know how to do this. I just tried to enter the table here using latex, but its not working. if someone could please explain how i can find the kernel by looking at the character table.

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Hint: Have you yet seen a proof that when $\chi$ is a character of a finite group $G,$ then for each $g \in G$ we have $|\chi(g)| \leq \chi(1),$ with equality if and only if .....? –  Geoff Robinson Jan 12 '12 at 14:17
Geoff is alluding to the fact that, $g$ belongs to the kernel iff $\chi(g)=\chi(1)$. So the kernel is the union of the conjugacy classes corresponding to the $\chi$-values which match the first column of the table. It's also interesting to note that every normal subgroup can be found from the table since every normal subgroup is the intersection of kernels of irred. reps. –  Bill Cook Jan 12 '12 at 14:30
–  Bill Cook Jan 12 '12 at 14:33
thank you so much Bill, I think I get it now. Geoff, I have come across the theorem and the proof, however I couldn't figure out how to use the table. Please tell me if I am correct in my thinking: if the first column for $\chi_1$=1 then all other g in that row which 1 as their value will be in the ker{$\chi_1$) –  sarah jamal Jan 12 '12 at 18:28
You could have asked this together with your question about centres, really. Also: this may sound obvious... but you could have asked this to whomever gave you the character table, probably! –  Mariano Suárez-Alvarez Jan 12 '12 at 19:09