When teaching the Higgs Mechanism it is useful to present the abelian case first. The mexican hat potential shows clearly that the ground state is not invariant under U(1), which is the spontaneously broken symmetry. At the same time one also sees that there are infinitely many ground states, all linked by U(1) transformations.
This picture is however often shown in relation to SSB in the SM. I think this is misleading, since now it is the gauge group SU(2)xU(1) to be spontaneously broken down to U(1). From the usual mexican hat potential figure, one cannot visualize this SSB correctly. Am I wrong?