Various questions have been posted on MO (some answered, some not) involving the character table of a finite group $G$ over a splitting field such as $\mathbb{C}$ of characteristic 0. My basic question is more about references:
1) Are there textbook or other convenient sources summarizing properties of $G$ which can or can't be deduced from a knowledge of its character table?
By itself this is a rather artificial question, since one doesn't usually know a character table without already knowing a lot about the group, but it provides good exercise material. Recently I was going over some standard theory with a graduate student and came across old notes from a course I taught decades ago, but I can't recall which books I consulted at the time.
Two sorts of information are typically deduced from a character table using the orthogonality relations: (a) numerical data, such as the order of $G$ or more generally all orders of centralizers and hence classes; (b) normal subgroup data, starting with the fact that normal subgroups are intersections of the kernels of characters and then deducing orders and inclusions of such subgroups. In particular, one can pinpoint the center and derived group, as well as determine whether or not $G$ is simple. On the other hand, it's well known that nonisomorphic groups can have the same character table (e.g., the two nonabelian groups of order 8); in particular, the character table can fail to predict the orders of the class representatives labelling columns.
One substantive question of this type which I'm unclear about is this:
2) To what extent does the character table determine properties of $G$ ranging from solvability to nilpotency?
Best Answer
For nilpotency, you can deduce the character table of $G/Z$ from the character table of $G$. First, determine $Z$. Second, throw out all the representations where $Z$ is not in the kernel. Third, merge the conjugacy classes which have the same trace in every representation. (This works because the irreducible representations of $G/Z$ are exactly the irreducible representations of $G$ with kernel containing $Z$, and because the inverse images of the conjugacy classes of $G/Z$ in $G$ are unions of conjugacy classes of $G$.) Then iterate.