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