We present an overview of the theory of finite groups, with regard to their application as flavour symmetries in particle physics. In a general part, we discuss useful theorems concerning group structure, conjugacy classes, representations and character tables. In a specialized part, we attempt to give a fairly comprehensive review of finite subgroups of SO(3) and SU (3), in which we apply and illustrate the general theory. Moreover, we also provide a concise description of the symmetric and alternating groups and comment on the relationship between finite subgroups of U (3) and finite subgroups of SU (3). Though in this review we give a detailed description of a wide range of finite groups, the main focus is on the methods which allow the exploration of their different aspects. *

We use the SmallGroups Library to find the finite subgroups of U (3) of order smaller than 512 which possess a faithful three-dimensional irreducible representation. From the resulting list of groups we extract those groups that can not be written as direct products with cyclic groups. These groups are important building blocks for models based on finite subgroups of U (3). All resulting finite subgroups of SU (3) can be identified using the well known list of finite subgroups of SU (3) derived by Miller, Blichfeldt and Dickson at the beginning of the 20 th century. Furthermore we prove a theorem which allows to construct infinite series of finite subgroups of U (3) from a special type of finite subgroups of U (3). This theorem is used to construct some new series of finite subgroups of U (3). The first members of these series can be found in the derived list of finite subgroups of U (3) of order smaller than 512. In the last part of this work we analyse some interesting finite subgroups of U (3), especially the group S 4 (2) ∼ = A 4 ⋊ Z 4 , which is closely related to the important Indeed a much larger "jump" occurs at g = 512: N(511) = 91774, while there are 10494213 groups of order 512 [19] of which only 30 are Abelian 3 . If we want to analyse groups with faithful 3-dimensional irreducible representations only, we don't need to consider groups of order 512 due to the following theorem:II.1 Theorem. Let D be an irreducible representation of a finite group G, then the dimension dim(D) of D is a divisor of the order ord(G) of G.The proof of this theorem can be found in textbooks on finite group theory, see for example [22] p. 176f. or [23] p. 288f. Note that theorem II.1 tells us that the order of any 3 The number of non-isomorphic Abelian groups of a given order can be calculated explicitly. See for example the article "Abelian Group" in [21].

We attempt to give a complete description of the "exceptional" finite subgroups Σ(36×3), Σ(72×3) and Σ(216×3) of SU (3), with the aim to make them amenable to model building for fermion masses and mixing. The information on these groups which we derive contains conjugacy classes, proper normal subgroups, irreducible representations, character tables and tensor products of their three-dimensional irreducible representations. We show that, for these three exceptional groups, usage of their principal series, i.e. ascending chains of normal subgroups, greatly facilitates the computations and illuminates the relationship between the groups. As a preparation and testing ground for the usage of principal series, we study first the dihedral-like groups ∆(27) and ∆(54) because both are members of the principal series of the three groups discussed in the paper. *

We determine the symmetry groups under which the charged-lepton and the Majorana-neutrino mass terms are invariant. We note that those two groups always exist trivially, i.e. independently of the presence of any symmetries in the Lagrangian, and that they always have the same form. Using this insight, we reevaluate the recent claim that, whenever lepton mixing is tri-bimaximal, S 4 is the minimal unique horizontal-symmetry group of the Lagrangian of the lepton sector, with S 4 being determined by the symmetries of the lepton mass matrices. We discuss two models for tri-bimaximal mixing which serve as counterexamples to this claim. With these two models and some group-theoretical arguments we illustrate that there is no compelling reason for the uniqueness of S

We propose a simple mechanism which enforces |U µj | = |U τ j | ∀j = 1, 2, 3 in the lepton mixing matrix U . This implies maximal atmospheric neutrino mixing and a maximal CP-violating phase but does not constrain the reactor mixing angle θ 13 . We implement the proposed mechanism in two renormalizable seesaw models which have features strongly resembling those of models based on a flavour symmetry group ∆(27). Among the predictions of the models, there is a determination, although ambiguous, of the absolute neutrino mass scale, and a stringent correlation between the absolute neutrino mass scale and the effective Majorana mass in neutrinoless double-beta decay. *