On a possible relationship between lepton mixing and the stability of dark matter

I comment on the proposal that the stability of dark matter may be due to an unbroken Z_2 symmetry contained in the partially broken lepton flavour symmetry group. I remark that (1) there is no Z_2 symmetry apparent in the lepton mass spectrum and in lepton mixing, (2) predictive models of this type may be constructed by using a lepton flavour symmetry group with three inequivalent singlets, to which the three left-handed-lepton gauge-SU(2) doublets are assigned, and (3) some predictions for the lepton masses and mixings are likely to be altered by radiative contributions to the neutrino mass matrix. I construct two models of this type in which the conserved Z_2 originates in a lepton flavour symmetry group D_4.