Representation Theory of Finite Groups [Lecture notes] by Anupam Singh

By Anupam Singh

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But center is always a normal subgroup which implies G is not simple. 3 (Burnside’s Theorem). Every group of order pa q b , where p, q are distinct primes, is solvable. Proof. We use induction on a + b. If a + b = 1 then G is a p-group and hence G is solvable. Now assume a + b ≥ 2 and any group of order pr q s with r + s < a + b is solvable. Let Q be a Sylow q-subgroup of G. If Q = {e} then b = 0 and G is a p-group and hence solvable. So let us assume Q is nontrivial. Since Q is a q-group (prime power order) it has nontrivial center.

1). We prove that, in fact, they form an orthonormal basis of H and generate as an algebra whole of C[G]. 5. The irreducible characters of G form an orthonormal basis of H, the space of class functions. 6. Let f ∈ H be a class function on G. Let (ρ, V ) be an irreducible representation of G of degree n with character χ. Let us define ρf = t∈G f (t)ρ(t) ∈ End(V ). Id where λ = |G| n f, χ . Proof. We claim that ρf is a G-map and use Schur’s Lemma to prove the result. For any g ∈ G we have, ρ(g)ρf ρ(g −1 ) = f (t)ρ(g)ρ(t)ρ(g −1 ) = t∈G f (t)ρ(gtg −1 ) = t∈G f (g −1 sg)ρ(s) = ρf .

Note that even if (ρ, V ) is an irreducible representation, (ρ|H , V ) need not be irreducible. The following theorem shows the intimate connection between the characters of a group G and any of its subgroup H. 3. INDEX TWO SUBGROUPS 47 define an inner product on C[G] denoted as , . We denote this inner product on C[H] by , H thinking of H as a group in itself. 3. Let G be a group and H its subgroup. Let ψ be a non zero character of H. Then there exists an irreducible character χ of G such that χ|H , ψ H = 0 where , H represents inner product on C[H] as explained above.

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