  We study the topological structure of the automorphism groups of compact
quantum groups showing that, in parallel to a classical result due to Iwasawa,
the connected component of identity of the automorphism group and of the
"inner" automorphism group coincide.
  For compact matrix quantum groups, which can be thought of as quantum
analogues of compact Lie groups, we prove that the inner automorphism group is
a compact Lie group and the outer automorphism group is discrete. Applications
of this to the study of group actions on compact quantum groups are
highlighted.
  We end with the construction of a compact matrix quantum group whose fusion
ring is not finitely generated, unlike the classical case.
