An exact derivation is given of the magnetic ground states for spin Ising modelswith pure four-spin interactions on the cubic lattices. The ordered states encompassferromagnetic, antiferromagnetic, and ferrimagnetic degenerate components, and theorder-parameter dimensionality is n=8, 4, and for the sc, the bcc, and the fcc lattices. The Landau-Ginzburg-Wilson Hamiltonians are derived for the sc and bcc lattices. Monte Carlo calculations demonstrate that the phase transition in all three lattices is of first order. The effects of a symmetry-breaking field are investigated for the bcc lattice. The phase diagram is calculated and shown to include lines of first-order and continuoustransitions as well as critical end points. The results are compared with mean-field and renormalization-group predictions.