TITLE: Problem 5.15, I. Martin Isaacs' Character Theory
QUESTION [1 upvotes]: Isaac's Character theory of finite groups book, Problem 5.15:
Let $H \subseteq G$ and suppose $\phi$ is a character of $H$ with $det(\phi)=1_{H}$. Let 
$\chi={\phi}^{G}$ and show ${(det\chi)}^{2}=1_{G}$.
Any idea?!

REPLY [0 votes]: Hint: Can you write down the block matrix of an induced representation? In your case, the given condition says that the determinant of any matrix in the image of the representation $\rho_{\phi}: H \to GL(V)$ is 1. If you try to write down the block matrix for the induced representation, you will see that up to permutation of rows, the matrix is a diagonal block matrix, each block has determinant 1. This gives the result.
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