We perform a systematic study of dihedral groups used as flavor symmetry. The key feature here is the fact that we do not allow the dihedral groups to be broken in an arbitrary way, but in all cases some (non-trivial) subgroup has to be preserved. In this way we arrive at only five possible (Dirac) mass matrix structures which can arise, if we require that the matrix has to have a non-vanishing determinant and that at least two of the three generations of left-handed (conjugate) fermions are placed into an irreducible two-dimensional representation of the flavor group. We show that there is no difference between the mass matrix structures for single-and double-valued dihedral groups. Furthermore, we comment on possible forms of Majorana mass matrices. As a first application we find a way to express the Cabibbo angle, i.e. the CKM matrix element |Vus|, in terms of group theory quantities only, the group index n, the representation index j and the index m u,d of the different preserved subgroups in the up and down quark sector: |Vus| =˛cos " π (mu−m d ) j n "˛w hich is | cos( 3 π 7 )| ≈ 0.2225 for n = 7, j = 1, mu = 3 and m d = 0. We prove that two successful models which lead to maximal atmospheric mixing and vanishing θ13 in the lepton sector are based on the fact that the flavor symmetry is broken in the charged lepton, Dirac neutrino and Majorana neutrino sector down to different preserved subgroups whose mismatch results in the prediction of these mixing angles. This also demonstrates the power of preserved subgroups in connection with the prediction of mixing angles in the quark as well as in the lepton sector.

It has recently been speculated that the solar neutrino mixing angle is connected to the golden ratio ϕ. Two such proposals have been made, cot θ 12 = ϕ and cos θ 12 = ϕ/2. We compare these Ansätze and discuss a model leading to cos θ 12 = ϕ/2 based on the dihedral group D 10 . This symmetry is a natural candidate because the angle in the expression cos θ 12 = ϕ/2 is simply π/5, or 36 degrees. This is the exterior angle of a decagon and D 10 is its rotational symmetry group. We also estimate radiative corrections to the golden ratio predictions. * 1 Actually, prediction (A) would lie very slightly outside the 2σ range of Ref. [3], which is sin 2 θ 12 = 0.278 ÷ 0.352.

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We discuss the possibility of obtaining a non-abelian discrete flavor symmetry from an underlying continuous, possibly gauged, flavor symmetry SU (2) or SU (3) through spontaneous symmetry breaking. We consider all possible cases, where the continuous symmetry is broken by small representations. "Small" representations are these which couple at leading order to the Standard Model fermions transforming as twoor three-dimensional representations of the flavor group. We find that, given this limited representation content, the only non-abelian discrete group which can arise as a residual symmetry is the quaternion group D ′ 2 .