  For a locally compact quantum group $\mathbb{G}$ we define its center,
$\mathscr{Z}(\mathbb{G})$, and its quantum group of inner automorphisms,
$\mathrm{Inn}(\mathbb{G})$. We show that one obtains a natural isomorphism
between $\mathrm{Inn}(\mathbb{G})$ and $\mathbb{G}/\!\mathscr{Z}(\mathbb{G})$,
we characterize normal quantum subgroups of a compact quantum group as those
left invariant by the action of the quantum group of inner automorphisms and
discuss several examples.
