  The key-scheduling algorithm in the AES is the component responsible for
selecting from the master key the sequence of round keys to be xor-ed to the
partially encrypted state at each iteration. We consider here the group
$\Gamma$ generated by the action of the AES-128 key-scheduling operation, and
we prove that the smallest group containing $\Gamma$ and all the translations
of the message space is primitive. As a consequence, we obtain that no proper
and non-trivial subspace can be invariant under its action.
