Lecture notes for Introduction to Representation Theory

⇒
⇒
Now pick V N such that ν V (g) ⇒= 0; it exists by Lemma 4.24. Theorem 4.21 implies that g
(and hence any element of C) acts by a scalar in V . Now let H be the subgroup of G generated
by elements ab −1 , a, b C. It is normal and acts trivially in V , so H = G, as V is nontrivial. Also
H = ⇒ 1, since | C | > 1.
Proof of Burnside’s theorem.
Assume Burnside’s theorem is false. Then there exists a nonsolvable group G of order p a q b . Let
G be the smallest such group. Then G is simple, and by Theorem 4.23, it cannot have a conjugacy
class of order p k or q k , k ⊂ 1. So the order of any conjugacy class in G is either 1 or is divisible
by pq. Adding the orders of conjugacy classes and equating the sum to paq b , we see that there has
to be more than one conjugacy class consisting just of one element. So G has a nontrivial center,
which gives a contradiction.
4.6 Representations of products
Theorem 4.25. Let G, H be finite groups, {V i } be the irreducible representations of G over a
field k (of any characteristic), and {W j } be the irreducible representations of H over k. Then the
irreducible representations of G × H over k are {V i W j }.
Proof. This follows from Theorem 2.26.
4.7 Virtual representations
Definition 4.26. A virtual representation of a finite group G is an integer linear combination of
irreducible representations of G, V = ⎨ n i V i , n i Z (i.e., n i are not assumed to be nonnegative).
The character of V is ν V := ⎨ n i ν Vi .
The following lemma is often very useful (and will be used several times below).
Lemma 4.27. Let V be a virtual representation with character ν V . If (ν V , ν V ) = 1 and ν V (1) > 0
then ν V is a character of an irreducible representation of G.
Proof. Let V 1 , V 2 , . . . , V m be the irreducible representations of G, and V = ⎨ n i V i . Then by
orthonormality of characters, (ν
2 2
V , ν V ) = ⎨ i . So ⎨ i n
i n i = 1, meaning that n i = ±1 for exactly
one i, and n j = 0 for j = i. But ν V (1) > 0, so n i = +1 and we are done.
4.8 Induced Representations
Given a representation V of a group G and a subgroup H → G, there is a natural way to construct
a representation of H. The restricted representation of V to H, ResG H V is the representation given
by the vector space V and the action δ Res G V = δ V | H .
H
There is also a natural, but more complicated way to construct a representation of a group G
given a representation V of its subgroup H.
Definition 4.28. If G is a group, H → G, and V is a representation of H, then the induced
representation Ind G H
V is the representation of G with
Ind G f(hx) = δ V (h)f(x)⊕x G, h H}
H V = {f : G ⊃ V | 54