Some questions we left open last time:

- Let be given by for some -th root of unity . Show that is . Conclude that are not isomorphic.
- Let be an matrix of finite order ( for some ). Show that , without using eigenvalues.
- In the proof of Prop. 1 of Serre (see Section 2.1), he states that “… can be defined to be a unitary matrix”. Show that every representation of a finite group is unitary. (As part of this exercise, give an appropriate definition of a unitary representation.)
- Prove that the character of is given by . If is a representation, prove that the character of a dual representation is . Find the characters of and .
- If is a map of representations, show that the kernel and image of are subrepresentations of respectively.
- Midterm question: Give your own proof of Serre Thm. 3 (section 2.3), that characters of irreducible representations form an orthonormal system.
- Prove Prop. 6 of Serre (section 2.5): If is an irreducible representation on of degree and character , and is a class function on , then is a homothety of ratio .
- Prove that the number of degree 1 representations on a finite group is .

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