  We investigate the normal subgroups of the groups of invertibles and
unitaries in the connected component of the identity. By relating normal
subgroups to closed two-sided ideals we obtain a "sandwich condition"
describing all the closed normal subgroups both in the invertible and in the
the unitary case. We use this to prove a conjecture by Elliott and Rordam: in a
simple C*-algebra, the group of approximately inner automorphisms induced by
unitaries in the connected component of the identity is topologically simple.
Turning to non-closed subgroups, we show, among other things, that in simple
unital C*-algebra the commutator subgroup of the group of invertibles in the
connected component of the identity is a simple group modulo its center. A
similar result holds for unitaries under a mild extra assumption.
