Character Theory of Finite Groups —Exercise Sheet 5 TU Kaiserslautern Jun.-Prof.Dr.Caroline Lassueur Dr.Inga Schwabrow Due date: Wednesday 21.06.2017, 14:00 SS 2017

Throughout this exercise sheet G denotes a finite group, V a C-vector space, and all vector spaces are assumed to be finite-dimensional. Each Exercise is worth 4 points.

Exercise 17 Let G be a finite group of odd order, and let r denote the number of conjugacy classes of G. Use character theory to prove that r G (mod 16). ≡ | | [Hint: Consider dual characters in your labelling of Irr(G).]

Exercise 18 Let A be a subring of a commutative ring B with 1. An element b B is called integral over ∈ A iff b is the zero of a monic polynomial f A[X]. In case B = C, then we say that b C is ∈ ∈ an algebraic integer if b is integral over A = Z.

(a) If b Q is an algebraic integer, then b Z. ∈ ∈ (b) If b B, then TFAE: ∈ (i) b is integral over A; (ii) A[b] is an A-module of finite type; (iii) there is a subring S of B containing A and b which is of finite type as an A-module.

(c) The set b B b is integral over A is a subring of B. { ∈ | }

Exercise 19 Prove that:

(a) \ Z(χ) = Z(G) . χ Irr(G) ∈

(b) If χ Irr(G), then χ(1) G : Z(χ) . ∈ | |

Exercise 20 (The determinant of a representation) If ρ : G GL(V) is a C-representation of G and det : GL(V) C denotes the determi- −→ −→ ∗ nant homomorphism, then we define

det := det ρ : G C∗ . ρ ◦ −→ Prove the following assertions:

(a) detρ is a linear character of G. (b) If G is a non-abelian simple group (or more generally if G is perfect, i.e. G = [G, G]), then the image ρ(G) of ρ is a subgroup of SL(V).

Turn the page over! (c) If G is a non-abelian simple group and χ Irr(G), then χ(1) , 2. ∈ (d) The finite groups D8 and Q8 cannot be distinguished by their character tables, but they can be distinguished by considering the determinants of their irreducible C- representations1.

Recall: D8 is the group of isometries of the square and admits the following presentation

4 2 1 1 D8 = σ, ρ ρ = σ = 1 and σρσ− = ρ− , h | i whereas Q8 is the quaternion group of order 8: as a set Q8 = 1, i j, k and is endowed {± ± ± ± } with the multiplication given by

2 2 2 2 1 = 1Q , ( 1) = 1 , i = j = k = 1 , i j = k = j i , j k = i = k j , k i = j = i k . 8 − − · − · · − · · − ·

1 Advice: If you work in groups of two students, then one student could compute the character table of D8 and the other the character table of Q8.