Historically, finite groups appeared in mathematics together with an action on some object, for example, as the groups of symmetries of polygons or polytopes. We are going to study, in some sense, the easiest possible actions of finite groups, namely, the linear actions on vector spaces. Such actions are called representations of finite groups. We will only consider representations in complex vector spaces.

Preliminary program:

1. Basic definitions and operations: direct sum, tensor product, dual.
2. Examples: some particular groups of small order, abelian groups (without proof of the completeness of the list so far).
3. Irreducible representations, complete reducibility theorem, proof using projections.
4. Unitary representations, second proof of complete reducibility theorem.
5. Functions on the group, orthogonality properties of matrix elements of representation, characters.
6. Applications of character theory.
7. Divisibility properties of dimensions.
8. Representation theory of symmetric groups.
9. Induced representations.
10. A criterium for irreducibility of an induced representation.