Exercise 4, § 3

Let $n$ be an integer $\geq 1$.

a) Show that the group $\mathbf{S}\mathbf{U}(n,\mathbf{C})$ is simply-connected (use Exerc. 2).

b) Show that the centre of $\mathbf{S}\mathbf{U}(n,\mathbf{C})$ consists of the matrices $\lambda .I_n$ for $\lambda \in \mathbf{C}$, $\lambda^n= 1$.

c) Every almost simple compact Lie group of type $A_n$ is isomorphic to the quotient of $\mathbf{S}\mathbf{U}(n+ 1,\mathbf{C})$ by the cyclic subgroup consisting of the matrices $\zeta^k.I_{n+1}(0\leq k < d)$, where $d$ divides $n+ 1$ and $\zeta$ is a primitive $d$th root of unity.

d) Prove that the group $\mathbf{S}\mathbf{L}(n,\mathbf{C})$ is simply-connected (use Chap. III, §6, no. 9, Th. 6).