Is it possible to determine if a group is simple by only looking at the character table?
My idea was to find the order of the group by summing the squares of the first column and then using Burnside's theorem. But of course, this works for a very special case. Alternately, I know that given a normal subgroup $N$ in $G$, an irreducible representation of $G/N$ can be extended to an irreducible representation of $G$. Perhaps this may be of use, since normal subgroups are union of conjugacy classes?