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 Un1 (C) × · · · × Unk (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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