  Product group unification is an attractive alternative to simple grand
unification. It solves the infamous doublet-triplet splitting problem and the
dimension-5 proton decay problems without introducing any fine-tuning.
Furthermore, the matter multiplets are still embedded into unified SU(5)
representations. In this paper, we discuss proton decay of the simplest product
group unification model based on SU(5)XU(2)_H . We find that the minimal setup
of the model has already been excluded by dimension-6 proton decay. We also
show that a simple extension of the model, with naturally generated SU(5)
incomplete multiplets, can rectify this problem. We find that the proton
lifetime will be in reach of coming experiments like DUNE and Hyper-K, when the
mass of the incomplete multiplet is associated with the Peccei-Quinn symmetry
breaking. In this case, the dark matter may be an admixture of the Wino LSP and
the axion.
