What's the difference between SU(N)xU(1) and U(N)? They seem to have the same # of generators: indeed, their generators seem to be exactly the same. The Lie algebra of their generators seem to be the same too. But I'm guessing they're not the same, since we call the Standard Model (before spontaneous symmetry breaking) SU(2)xU(1) and not U(2). But I don't know how they're different mathematically.