Neutrino masses, mixing, and oscillations in the S4 model of permutation symmetry

In the ${\mathrm{S}}_4$ model, we work out the simplest possible scheme for neutrino masses. We find that ${\nu }_e$ and ${\nu }_{\mu }$ have dominant massless components but not ${\nu }_{\tau }$. The mixing matrix is completely determined and ${\nu }_e-{\nu }_{\tau }$ and ${\nu }_{\mu }-{\nu }_{\tau }$ oscillations are possible with known mixing angles and only one $\delta m^2$.