Homework 2 Due Friday, February 2nd You may find all the Exercises as attached. 1. Let Sn be the symmetric group. Construct a nontrivial one-dimensional representation ρ : Sn −→ GL1 (R) = R× . 2. Recall the dihedral group Dn generated by the elements r and s, satisfying the relations rn = 1,

s2 = 1,

srs = r−1 .

Suppose that n is even. Find all the one dimensional complex representations of Dn . 3. Let ρ : SU2 −→ C× be a continuous group homomorphism. Prove that ∀g ∈ SU2 ,

ρ(g) = 1.

4. Recall the unitary group U1 = {z ∈ C | |z| = 1}. Find all the one dimensional continuous complex representations of U1 . 5. Let G be a finite group. Let ρ : G −→ GLn (R) be an n-dimensional real representation. Prove that there exists a matrix τ ∈ GLn (R) such that ∀g ∈ G,

τ ρ(g)τ −1 ∈ On (R).

In other words, ρ is isomorphic to an orthogonal representation. 6. Let G be a finite group. Suppose that ρ : G −→ SL2 (R) is an injective group homomorphism. Prove that G is a cyclic group.