Exercise 11, § 4

(a) For $x, y$ in $S_{2n-1} \subset \mathbf{C}^n$, we define $\alpha(x, y) \in (0, \pi)$ by

For s, t in $\mathbf{U}(n)$, we write $d(s, t) = \sup_{x \in S_{2n-1}} \alpha(sx, tx)$. Show that d is a left and right invariant distance on $\mathbf{U}(n)$.

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

(c) Let s, t be in $\mathbf{U}(n)$ be such that $\theta(t) < \frac{\pi}{2}$ and s commutes with (s, t). Then s and t commute. (In the notation of (b), let $V_j'$ be the orthogonal supplement of $V_j$ and $W_j = t(V_j)$; show that $W_j = (W_j \cap V_j) + (W_j \cap V_j')$ and then that $W_j \cap V_j' = \{0\}$.)

(d) Show that there exists a compact neighbourhood V of e in $\mathbf{U}(n)$ which has the property of Exercise 10, which is stable under inner automorphisms of $\mathbf{U}(n)$ and which is such that if $x, y$ are non-permutable elements of $V$ then $x$ and $(x, y)$ are not 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 $\beta(V_1) > 1/p$. Show that, for every finite subgroup $F$ of $\mathbf{GL}(n, \mathbf{C})$, there exists a commutative normal subgroup $A$ of $F$ such that $\mathrm{Card}(F/A) \leq p$ (Jordan’s theorem). (Reduce it to the case where $F \subset \mathbf{U}(n)$. Take $A$ to be the subgroup of $F$ generated by $F \cap V$ and use (d).)