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?

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- May 11th 2011, 12:46 PMTurloughmackCharacter 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? - May 11th 2011, 05:30 PMDrexel28
Think back to what irreducible characters are--they are the characters of irreps on $\displaystyle G$. So, let $\displaystyle \chi\in\text{irr}(G)$ and let $\displaystyle \rho:G\to\text{GL}\left(V\right)$ be the irrep it came from. Then, as you can easily verify $\displaystyle \chi^a$ as you denoted is the character of $\displaystyle \rho\circ a:G\to\text{GL}(V)$ (which is a representation since $\displaystyle a\in\text{End}(G)$). Now, if $\displaystyle \rho\circ a$ left any non-trivial proper subspace of $\displaystyle V$ $\displaystyle \rho\circ a$-invariant wouldn't it be necessary (using the fact that $\displaystyle a$ is bijective) that same subspace would be $\displaystyle \rho$-invariant? So...