It is very well-known that, if $G$ is a finite group, then the (complex) characters of $G$ separate the conjugacy classes; that is, if $g, h \in G$ are not conjugate, then there exists a character $\chi$ such that $\chi(g) \ne \chi(h)$. However, I have been wondering about the minimal number of characters needed to separate all the classes.


Consider the (very basic) example $G=C\_n$, a cyclic group of order $n$. Then it is possible to find a *single* character $\chi$ which is injective on $G$, so the answer is $1$ in this case.


Looking at a few character tables using GAP, it seems the answer is often 1.


Is there any literature on this? Any reason why this number could be much easier to determine than I think? Is it always 1 ? I may very well have overlooked something simple.


Note that the answer would be different if you only allowed irreducible characters (then for $C\_2\times C\_2$, it is 2); it's also different if you allow virtual characters, I guess.


Thanks!


Pierre



number of characters needed to separate conjugacy classes
You always find a character which distinguishes between the distinct conjugacy classes. Let $\delta\_k$ be the characteristic function of the $k$-th conjugacy class. Set $f=\delta\_1+2\delta\_2+3\delta\_3+\dots$. Then clearly $f$ distinguishes the conjugacy classes. As a class function, $f$ is a complex linear combination of the irreducible characters. Write $f=a+ib$, where $a$ and $b$ are real linear combinations of irreducible characters. Then, for a suitable real number $\lambda$, $g=a+\lambda b$ also distinguishes the conjugacy classes. Now $g$ is a real linear combination of the irreducibles. Replacing the real coefficients by sufficiently good rational approximations, we still have the separating property. Now multiply by an integer to obtain a virtual character, and finally add a suitably large multiple of the regular character to obtain the required character.


*Note added:* The question actually doesn't have much to do with character theory, since the following far more general fact holds: If in a complex matrix there are no two equal columns, then there is a nonnegative integral linear combination of the rows such that the entries are pairwise distinct. This can for instance be proven by induction on the number of columns.


