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We present a theory of spin, electronic, and transport properties of a few-electron lateral triangular triple quantum dot molecule in a magnetic field. Our theory is based on a generalization of a Hubbard model and the linear combination of harmonic orbitals combined with configuration interaction method for arbitrary magnetic fields. The few-particle spectra obtained as a function of the magnetic field exhibit Aharonov-Bohm oscillations. As a result, by changing the magnetic field, it is possible to engineer the degeneracies of single-particle levels, and thus, control the total spin of the many-electron system. For the triple dot with two and four electrons, we find oscillations of total spin due to the singlet-triplet transitions occurring periodically in the magnetic field. In the three-electron system, we find a transition from a magnetically frustrated to a spin-polarized state. We discuss the impact of these phase transitions on the …