Characters of Normal Subgroup of Index 2

[Ant farm]

Hi There,

Ok, I'm new to this so I'm sorry if this is abit warbled!...

We have a normal subgroup N of a finite group H such that [H:N]=2
We have a chatacter Chi belonging to the irreducible characters of H, Irr(H) , which is zero on H\N.

I have already shown that Chi restricted to N = Theta 1 + Theta 2, Where
Theta 1 is not equal to Theta 2, and both Theta 1, and Theta 2 belong to Irr(N).

I am now trying to show that given a conjugacy class K of H which is also a conjugacy class of N, we get Theta 1 (n) = Theta 2 (n) for n a member of K.

Intuitively, I can see this is true, I have seen examples in the alternating group A6, the normal subgroup of index 2 in S6 fro example.

I would be really greatful for a push in the right direction...I have books: 'James and Liebeck' and 'Isaacs'... if you could even direct me to some relevant info?

Wow, think my notation is abit crap, sorry.

 

[matt grime]

It is traditional (though not necessarily ideal) to use ascii-tex to write maths in electronic form if you're not going to use the LaTeX capabilities of this site.

Thus \theta_1 is written as theta_1 (or \theta_1).

_ means subscript and ^ means superscript.

Right. I don't know if you're supposed to do it this way but here's my idea.

Consider Ind_N^H(theta_1).

 

[Ant farm]

Thanks for the notation tip, don't know how to use LaTeX so will try the other method.
Ok, I'm looking at Ind_N^H(theta_1), using Frobenius Reciprocity, I have <Ind_N^H(theta_1) , Chi> = <theta_1, Chi_N> (where < , > denotes inner product)

Since Chi_N = theta_1 + theta_2 we get <Ind_N^H(theta_1) , Chi> = <theta_1, theta_1 + theta_2>

now, is <theta_1, theta_1 + theta_2> = <theta_1, theta_1> + < theta_1, theta_2> ? and where can i go from here to get to my result...as in have theta_1(n) = theta_2(n) for each n belonging to the conjugacy class K

 

[matt grime]

It's an inner product - it is of course (conjugate) linear in each variable. You have possibly learned (and I was assuming you had) that irreducible characters form an orthonormal basis of the class functions. However, if you didn't know that <A+B,C>=<A,C>+<B,C> then I have perhaps made a mistake in where you've got to in learning the material.

To get the final result you just need to look at how to construct in induced character.