Exercise 11, § 4

(a) For x, y in $S_{2n-1} \subset \mathbf{C}^n$, define $\alpha(x, y) \in [0, \pi]$ by $\cos \alpha(x, y) = \Re((x|y))$. For s, t in $\mathbf{U}(n)$, set $d(s, t) = \sup_{x \in S_{2n-1}} \alpha(sx, tx)$. Prove that d is a distance on $\mathbf{U}(n)$ invariant on the left and on the right.

(b) Let $s \in \mathbf{U}(n)$. Let $\theta_1, \ldots, \theta_m$ be the numbers of $]-\pi, \pi]$ such that the $e^{i\theta_j}$ are the distinct eigenvalues of s. Put $\theta(s) = \sup_{1 \leq j \leq m} |\theta_j|$. Then $d(e, s) = \theta(s)$. (Let $V_j$ be the proper subspace of s corresponding to $e^{i\theta_j}$. For $x \in S_{2n-1}$, lower-bound $\Re((x|sx))$ by decomposing according to the $V_j$.)

(c) Let s, t in $\mathbf{U}(n)$ such that $\theta(t) < \frac{\pi}{2}$ and such that s commutes with (s, t). Then s and t commute.

(With the notations of b), let $V_j'$ be the orthogonal supplementary of $V_j$ and $W_j = t(V_j)$; prove that $W_j = (W_j \cap V_j) + (W_j \cap V_j')$, then that $W_j \cap V_j' = \{0\}$.)

(d) Prove that there exists a compact neighbourhood V of e in $\mathbf{U}(n)$ which has the property of exerc. 10, which is stable under the inner automorphisms of $\mathbf{U}(n)$, and which is such that if x, y are non-permutable elements of V, x and (x, y) are non-permutable. (Use c).)

(e) Let $\beta$ be the normalized Haar measure of $\mathbf{U}(n)$. Let $V_1$ be a symmetric compact neighbourhood of e in $\mathbf{U}(n)$ such that $V_1^2 \subset V$. Let $p$ be an integer such that $\beta(V_1) > 1/p$. Prove that, for every finite subgroup F of $\mathbf{GL}(n, \mathbf{C})$, there exists a normal commutative subgroup A of F such that Card(F/A) $\leq p$ (Jordan's theorem). (Reduce to the case where F $\subset \mathbf{U}(n)$. Take for A the subgroup of F generated by F $\cap$ V and use d).)