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Math Help - Character Theory II

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    Character Theory II

    Urgent answer required for these questions please!

    (3) Let G be a finite group and let element a in Aut(G) be an automorphism og G. Let X be an element of Irr(G) be an irreducible character of G.
    Define (X)^a(g) := X(g^a). Is (X)^a irreducible?
    Last edited by Ackbeet; May 11th 2011 at 02:00 PM. Reason: Splitting threads.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Turloughmack View Post
    Urgent answer required for these questions please!

    (3) Let G be a finite group and let element a in Aut(G) be an automorphism og G. Let X be an element of Irr(G) be an irreducible character of G.
    Define (X)^a(g) := X(g^a). Is (X)^a irreducible?
    Think back to what irreducible characters are--they are the characters of irreps on G. So, let \chi\in\text{irr}(G) and let \rho:G\to\text{GL}\left(V\right) be the irrep it came from. Then, as you can easily verify \chi^a as you denoted is the character of \rho\circ a:G\to\text{GL}(V) (which is a representation since a\in\text{End}(G)). Now, if \rho\circ a left any non-trivial proper subspace of V \rho\circ a-invariant wouldn't it be necessary (using the fact that a is bijective) that same subspace would be \rho-invariant? So...
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