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 . So, let and let be the irrep it came from. Then, as you can easily verify as you denoted is the character of (which is a representation since ). Now, if left any non-trivial proper subspace of -invariant wouldn't it be necessary (using the fact that is bijective) that same subspace would be -invariant? So...