Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Characters of Normal Subgroup of Index 2

  1. Jan 7, 2007 #1
    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.
  2. jcsd
  3. Jan 7, 2007 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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 [itex]\theta_1[/itex] 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).
  4. Jan 8, 2007 #3
    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
  5. Jan 8, 2007 #4

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    It's an inner product - it is of course (conjugate) linear in each variable. You have possibly learnt (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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook