I am following the text by Isaacs on character theory and I have a few questions.
From p. 10, it seems like an reducible representation is one whose matrix at each group element can be written in a block diagonal form. However, the proposition on p. 20 states that the matrix of a representation at every group element can be written as a diagonal matrix. This would imply that every representation is reducible, which is clearly nonsense. What am I missing here?