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The discrete non-Abelian symmetry A₄, valid at some high-energy scale, naturally leads to three degenerate neutrino masses, without spoiling the hierarchy of charged-lepton masses. Realistic neutrino mixing angles (one of which is necessarily maximal and the other large) are then automatically induced radiatively, and the correct mass splittings may be obtained in the context of softly broken supersymmetry. The quark mixing matrix is also calculable in a similar way. The mixing parameter U_e3 is predicted to be imaginary, leading to maximal CP violation in neutrino oscillations. Neutrinoless double beta decay and τ→μγ should be in the experimentally accessible range.