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.
Generalities on linear representations, irreducible representations and operation on representations. Character theory, Schur’s lemma, decompositions of representations. Subgroups and induced representations. Explicit examples. Representations of the symmetric group.