6 1. Introduction (2) (Discreteness) G is finite. (3) (Lie-type structure) G is a subgroup of Un(C) (the group of unitary n × n matrices) for some n. Then there is a subgroup G of G such that (i) (G is close to G) The index |G/G | of G in G is On(1) (i.e., bounded by Cn for some quantity Cn depending only on n). (ii) (Nilpotent-type structure) G is abelian. A key observation in the proof of Jordan’s theorem is that if two unitary elements g, h ∈ Un(C) are close to the identity, then their commutator [g, h] = g−1h−1gh is even closer to the identity (in, say, the operator norm op ). Indeed, since multiplication on the left or right by unitary elements does not affect the operator norm, we have [g, h] − 1 op = gh − hg op = (g − 1)(h − 1) − (h − 1)(g − 1) op and so by the triangle inequality (1.2) [g, h] − 1 op ≤ 2 g − 1 op h − 1 op . Now we can prove Jordan’s theorem. Proof. We induct on n, the case n = 1 being trivial. Suppose first that G contains a central element g (i.e., an element that commutes with every element in G) which is not a multiple of the identity. Then, by definition, G is contained in the centraliser Z(g) := {h ∈ Un(C) : gh = hg} of g, which by the spectral theorem is isomorphic to a product Un 1 (C) × · · · × Un k (C) of smaller unitary groups. Projecting G to each of these factor groups and applying the induction hypothesis, we obtain the claim. Thus we may assume that G contains no central elements other than multiples of the identity. Now pick a small ε 0 (one could take ε = 1 10n in fact) and consider the subgroup G of G generated by those elements of G that are within ε of the identity (in the operator norm). By considering a maximal ε-net of G we see that G has index at most On,ε(1) in G. By arguing as before, we may assume that G has no central elements other than multiples of the identity. If G consists only of multiples of the identity, then we are done. If not, take an element g of G that is not a multiple of the identity, and which is as close as possible to the identity (here is where we crucially use that G is finite). Note that g is within ε of the identity. By (1.2), we see that if ε is suﬃciently small depending on n, and if h is one of the generators of G , then [g, h] lies in G and is closer to the identity than g, and is thus a

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