Presentation: Representation Theory and Group Theory

The material below is a presentation I gave at the start of February in 2019 about the abelianness of groups of order p^{2} using representations.

Representation Theory and Group Theory

Representation theory of finite groups is a deep theory in its own right, but sometimes we can use representations to give some elegant proofs of statements that belong to group theory. Today I’m going to give an example of such a situation. We’re going to use some representation theory to prove the following proposition:

Proposition: Let G be a group of order p^{2} where p is a prime. Then every irreducible representation of G is one-dimensional. In particular, G is abelian.

Before we can give the proof, we’ll need the definition of a representation and give some simple facts.

Definition: If G is a group, F a field, and V a vector space over F, then a representation is a homomorphism G \xrightarrow{\varphi} GL(V). We sometimes write \varphi for brevity. A subrepresentation of \varphi is a representation G \xrightarrow{\varphi|_{W}} GL(W) with W a vector subspace of V and \varphi|_{W}(g) = \varphi(g)|_{W} for every g \in G. The representation \varphi is irreducible if it has no proper subrepresentation. The degree of \varphi is the dimension of V.

Facts: Suppose G is a finite group.

  1. If \varphi is an irreducible representation of G, then the degree of \varphi divides the order of G.
  2. The number of irreducible representations of G is equal to the number of conjugacy classes of G.
  3. Let k be the number of conjugacy classes of G. If n_{i} is the degree of the i-th irreducible representation, then |G| = \sum_{i = 1}^{k}n_{i}^{2}.

These facts are stated without proof. If you take any introductory class in representation theory, then you will very likely prove these facts within the first two weeks of class. They are fairly straightforward, but some other machinery needs to be developed in order to prove them. Now let us prove the proposition.

proof: Let \varphi be an irreducible representation of G. By fact (1), the degree of \varphi is either p^{2}, p, or 1. We will show it cannot be p^{2} or p. If the degree of \varphi is p^{2} then we contradict fact (3) since \sum_{i = 1}^{k}n_{i}^{2} \ge p^{4} > |G|. If the degree of \varphi is p, then fact (3) tells us that \varphi is the only irreducible representation of G. However the representation G \xrightarrow{\psi} GL(V), where V is a one-dimensional vector space, given by \psi(g) = 1 for all g \in G (also known as the trivial representation) is an irreducible representation of degree 1. So, \varphi cannot be the only irreducible representation and therefore the degree of \varphi cannot be p. This forces the degree of \varphi to be 1, and since \varphi was an arbitrary irreducible representation we conclude every irreducible representation of G is of degree 1. It follows by facts (3) and (2) that that the number of conjugacy classes of G is exactly the order of G. This happens if and only if G is abelian, so we are done.

There are also some more classical examples of using representation theory to prove facts about group theory. Two of the most common are proofs of Burnside’s theorem and of the Frobenius kernel being a normal subgroup. They mainly use character theory, a subtheory of representations. That concludes this presentation, thanks for reading!

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