This paper aims at providing a systematic treatment of the crystallographic point groups. Some well-known properties of them, in terms of the theory of the poles of finite rotations, are first discussed, so as to provide a simple way for recognizing their invariant subgroups. A definition of the semi-direct product is then given, and it is shown that all crystallographic point groups can be expressed as a semi-direct product of one of their invariant subgroups by a cyclic subgroup. Many useful relations between point groups can be obtained by exploiting the properties of the triple and mixed triple semi-direct products, which are defined. Much of the rest of the paper is devoted to the theory of the representations of semi-direct products. The treatment here parallels that given by Seitz (1936) for the reduction of space groups in terms of the representations of its invariant subgroups (the translation groups). The latter, however, are always Abelian and this is not always the case for point groups. The full treatment of the general case, such as given by McIntosh (1958), is laborious and it is shown that, if the emphasis is placed on the bases of the representations, rather than the representations themselves, it is possible to achieve the reduction of the point groups by a method hardly more involved than that required when the invariant subgroup is Abelian. It is also shown that, just as for space groups, the representations of the invariant subgroups can be denoted and visualized by means of a vector, which allows a very rapid classification of the representations, very much as the k vector as used by Bouckaert, Smoluchowski & Wigner (1936) allows the formalism of the Seitz method for space groups to be carried out in a graphical fashion. One of the major consequences of this work is that it affords a substantial simplification in the use of the symmetrizing and projection operators that are required to obtain symmetry-adapted functions: a very systematic alternative to the method given by Melvin (1956) is therefore provided. In the last section of the paper all the techniques discussed are applied in detail, as an example, to the cubic groups. The projection operators are used to obtain symmetry-adapted spherical harmonics for these groups. The paper might be found useful as an introduction to the methods for the reduction of space groups.
Invariants and semi-direct products for finite group actions on tensor categories
