This chapter concerns the numerical properties of character values, and how this can be used to prove results about representations and about finite groups. The matrices that represent group elements power to the identity, so their eigenvalues are roots of unity. This means that character values are algebraic integers, roots of monic polynomials with integer coefficients.

After an introductory section concerning some basic properties about algebraic integers we see some applications of algebraic integer theory to finite group representations. We first obtain a formula for the idempotent \(e\) in \(\mathbb{C} G\) associated to a given character, which is how we were able to produce the idempotents in Chapter 1.

We then prove that the degree of an irreducible character divides the order of the group. Currently there does not appear to be a proof of this fact that does not use algebraic integers, or some kind of number theory. A representation-theoretic proof would be of interest. Note that this is not true over fields of characteristic \(p\); the group \(\mathrm{SL}_2(p)\) has one irreducible representation over a field of characteristic \(p\) of each degree \(1,2,\dots,p\), but \(|\mathrm{SL}_2(p)|\) is not divisible by all these numbers.

The highlight of this chapter is Burnside's \(p^\alpha q^\beta\)-theorem, which asserts that all finite groups divisible by exactly two distinct primes are soluble. Although a group-theoretic proof of this result is known, it is considerably more difficult than this proof, which can be taught in an introductory course.

• 3.1 Algebraic integer preliminaries
• 3.2 Applications of central elements
• 3.3 Burnside's p^αq^β-theorem