Nonlinear Characters and Finite Groups: A Case Study

[Ant farm]

Hi,
Ok, I have notice that for several finite groups the following situation occurs... I will use the non-abelian group of order 27 to illustrate the point I'm making:

The group has 11 charachers, 9 of which are linear.
The group has derived subgroup G' (= Z(G) the centre of the group...irrelevant!) has 3 elements, G/G' is isomorphic to C_3 x C_3

If the non linear characters are called Chi_10 and Chi_11, why are they equal to zero on G/G'?

Another example of where this occurs would be A_4, which has 3 linear characters, and one non-llinear, and the derived subgroup= V_4 so Chi_4, the non-linear character = 0 on the conjugacy classes (123) and (132)

Thank You very much!

 

[matt grime]

This is an aspect of a named theory whose name escapes me... gah, that's annoying: the relation between characters of G and G/N for some normal subgroup. ARRRGH.

Anyway. Chi_10 cannot be zero on G/G' since Chi_10 is not a character of G/G'. It cannot afford a rep of G/G' unless G' is in the kernel of Chi_10. Anyway, the identity is certainly an element of G/G' so that can't be what you mean anyway.

Here's an idea for you to think about, that might be to do with what you're trying to get at. Take a (linear) character of G', now induce it up to G. What happens? Perhaps you might well be adding up w,w^2 and w^3 where w is a cube root of unity. What is w+w^2+w^3. Note: I have not checked this since I don't have pen or paper at hand. I might be selling you a dummy, sorry if I am.