Lecture notes for Introduction to Representation Theory

Now, consider
∂i ν V (g Ci ).
This is an algebraic integer, since:
(i) ∂ i are algebraic integers by Proposition 4.17,
i
(ii) ν V (g Ci ) is a sum of roots of unity (it is the sum of eigenvalues of the matrix of δ(g Ci ), and
|G |
since g C = e in G, the eigenvalues of δ(gCi ) are roots of unity), and
i
(iii) A is a ring (Proposition 4.12).
On the other hand, from the definition of ∂ i ,
∂i ν V (g Ci ) = |C i |ν V (g Ci )ν V (g Ci ) .
dim V
Recalling that ν V is a class function, this is equal to
C i
i
νV (g)ν V (g) |G|(ν V , ν V ) = .
dim V dim V
gG
Since V is an irreducible representation, (ν V , ν V ) = 1, so
∂i ν V (g Ci ) =
C i
|G|
.
dim V
|G|
Since dim
V
Q and ⎨ C i
∂ i ν V (g Ci ) A, by Proposition 4.13 dim V
Z.
|G|
4.5 Burnside’s Theorem
Definition 4.18. A group G is called solvable if there exists a series of nested normal subgroups
where G i+1 /G i is abelian for all 1 ∗ i ∗ n − 1.
{e} = G 1 γ G 2 γ . . . γ G n = G
Remark 4.19. Such groups are called solvable because they first arose as Galois groups of polynomial
equations which are solvable in radicals.
a
Theorem 4.20 (Burnside). Any group G of order p q b , where p and q are prime and a, b ⊂ 0, is
solvable.
This famous result in group theory was proved by the British mathematician William Burnside
in the early 20-th century, using representation theory (see [Cu]). Here is this proof, presented in
modern language.
Before proving Burnside’s theorem we will prove several other results which are of independent
interest.
Theorem 4.21. Let V be an irreducible representation of a finite group G and let C be a conjugacy
class of G with gcd(|C|, dim(V )) = 1. Then for any g C, either ν V (g) = 0 or g acts as a scalar
on V .
52