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] =
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
= gh hg
= (g 1)(h 1) (h 1)(g 1)
and so by the triangle inequality
(1.2) [g, h] 1
2 g 1
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 =
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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