Uniqueness of SU(5) and SO(10) grand unified theories

We will prove that SU(5) and SO(10) grand unified theories are almost uniquely chosen under a few reasonable assumptions. The main ansatz is that there exist only left-handed particles in an SU(2) doublet and their antiparticles in an SU(2) singlet. For the case of a single-multiplet unification, SO(10) is unique, using the facts that the charge of the d quark is not neutral and the theory should be anomaly free. For the case of two-multiplet unification, SU(5) is unique among SU(N) (N≤500). The color group is effectively determined as the standard SU(3) with triplets of colored quarks. Quark charges must be the standard 2/3 and -1/3. No unifying simple group exists for G(horizontal) ×G(GUT) under the assumptions above. For the case of SU(5), the cancellation of the triangle anomaly emerges as a consequence rather than a hypothesis.