This chapter introduces representations and modules for finite groups. We start with some constructions for vector spaces. We consider the direct sum, tensor product, dual space, and more generally \(\mathrm{Hom}_k(V,W)\), the vector space of all linear transformations between two vector spaces \(V\) and \(W\). The key point of the first section is 'tensor-Hom adjunction', a relationship between the space of homomorphisms and the tensor product. We prove a stronger version of the classic tensor-Hom adjunction, because in our situation there is a stronger version available. It will be vital in proving our results in character theory in Section 2.1.

We then consider the situation of so-called irreducible subgroups of the general linear group of invertible linear transformations on a vector space \(V\). Subgroups of \(\mathrm{GL}(V)\) that stabilize no proper, non-zero subspace are called 'irreducible', and the centralizer of these subgroups in \(\mathrm{GL}(V)\) consist purely of scalar matrices, at least if the vector space is over an algebraically closed field.

We move on to Wedderburn's theorem, which gives a classification of simple rings (no two-sided ideals) with a condition on their one-sided ideals. They will all be shown to be matrix rings over a division ring. This has an immediate application later in the chapter, when we look at group algebras.

In Section 1.4 we introduce two concepts that will stay with us for the rest of the course: algebras and modules. The basic object of study in group representation theory is group algebras, a ring that is associated to a group. This section proves the Jordan-Hölder theorem for modules and Schur's lemma, two basic results.

Afterwards we prove Maschke's theorem, which states that if a finite group \(G\) acts on a vector space \(V\) over a field of characteristic not dividing \(|G|\), and \(G\) stabilizes a subspace \(U\) of \(V\), then \(V\) stabilizes a complement \(W\) to \(U\) in \(V\), so a subspace such that \(U\cap W=0\) and \(U+W=V\). This distinguishes representation theory over the complex numbers from representation theory over fields of prime characteristic, and can be considered the most important result in representation theory. Here we combine Maschke's theorem and Wedderburn's theorem to break the group algebra up into pieces, each being a matrix algebra over a division ring.

We finally look back at the vector space constructions from the first section, turning each into a module over the group algebra.

This chapter consists of the basics of representation theory from a module point of view.

• 1.1 Some new vector spaces
• 1.2 Irreducible matrix groups
• 1.3 Wedderburn's theorem
• 1.4 Algebras and modules
• 1.5 Maschke's theorem
• 1.6 Homs and tensors