# Thread: Character Theory II

1. ## 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?

2. Originally Posted by Turloughmack
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...