John Stillwell - Naive Lie Theory.pdf - Index of

3.8 Connectedness and discreteness 69
Exercises
It happens that the quotient of each of the groups SO(n), U(n), SU(n), Sp(n) by
its center is a group with trivial center (see Exercise 3.8.1). However, it is not
generally true that the quotient of a group by its center has trivial center.
3.7.1 Find the center Z(G) of G = {1,−1,i,−i,j,−j,k,−k} and hence show that
G/Z(G) has nontrivial center.
3.7.2 Prove that U(n)/Z(U(n)) = SU(n)/Z(SU(n)).
3.7.3 Is SU(2)/Z(SU(2)) = SO(3)?
3.7.4 Using the relationship between U(n), Z(U(n)), andSU(n), orotherwise,
show that U(n) is path-connected.
3.8 Connectedness and discreteness
Finding the centers of SO(n), U(n), SU(n), andSp(n) is an important step
towards understanding which of these groups are simple. The center of
any group G is a normal subgroup of G, hence G cannot be simple unless
Z(G)={1}. This rules out all of the groups above except the SO(2m +1).
Deciding whether there are any other normal subgroups of SO(2m + 1)
hinges on the distinction between discrete and nondiscrete subgroups.
A subgroup H of a matrix Lie group G is called discrete if there is a
positive lower bound to the distance between any two members of H, the
distance between matrices (a ij ) and (b ij ) being defined as
√
∑ |a ij − b ij | 2 .
i, j
(We say more about the distance between matrices in the next chapter.) In
particular, any finite subgroup of G is discrete, so the centers of SO(n),
SU(n), andSp(n) are discrete. On the other hand, the center of U(n) is
clearly not discrete, because it includes elements arbitrarily close to the
identity matrix.
In finding the centers of SO(n),SU(n),andSp(n) we have in fact found
all their discrete normal subgroups, because of the following remarkable
theorem, due to Schreier [1925].
Centrality of discrete normal subgroups. If G is a path-connected matrix
Lie group with a discrete normal subgroup H, then H is contained in the
center Z(G) of G.