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.

 

[tacman]

Thank you for sharing this interesting observation. This is a common occurrence in finite groups, and it has to do with the structure of the group and the properties of its characters. In general, a finite group G of order n has n distinct irreducible characters, and these characters can be either linear or nonlinear. The number of linear characters is equal to the number of conjugacy classes in G, and the remaining characters are nonlinear.

In the case of the non-abelian group of order 27, we see that there are 11 characters, 9 of which are linear. This is because the group has 9 conjugacy classes, and each conjugacy class corresponds to a linear character. The remaining 2 characters, Chi_10 and Chi_11, are nonlinear.

Now, let's consider the derived subgroup G' of this group. As you mentioned, G' has 3 elements and G/G' is isomorphic to C_3 x C_3. This means that G/G' has 2 conjugacy classes, and therefore, only 2 linear characters. Since Chi_10 and Chi_11 are nonlinear characters, they are not defined on the elements of G/G' and therefore, their values are equal to zero on G/G'.

Similarly, in the case of A_4, we see that the derived subgroup V_4 has 2 elements, and A_4/V_4 is isomorphic to C_3. This means that there is only 1 conjugacy class in A_4/V_4, and therefore, only 1 linear character. Again, since Chi_4 is a nonlinear character, it is not defined on the elements of A_4/V_4, and its value is equal to zero on these elements.

In general, nonlinear characters are not defined on the elements of the derived subgroup, and therefore, their values are equal to zero on these elements. This is a consequence of the fact that nonlinear characters are not induced from linear characters, and they do not have a nice relationship with the structure of the group.

I hope this explanation helps to clarify the situation. Thank you for sharing your case study and bringing attention to this interesting phenomenon.