  By introducing a result that guarantees a given bialgebra to be a Hopf
algebra under a natural condition, we show that the quantum automorphism group
of the algebra k[x] of polynomials over a field k (of any characteristic) is
the universal quantum a x + b group, generalizing the fact that the
automorphism group of k[x] is the a x + b group. The q-deformation of the a x +
b group is then seen as one among a certain family ${\mathcal A}_{q, n}$ of
quantum subgroups of this universal quantum group.
