Construction of the symmetry groups of polymer molecules

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Abstract
A derivation of the line groups, which are the symmetry groups of the stereo-regular polymer molecules, is presented. Every line group is an extension of a one-dimensional translation group by a point group. The point group is either cyclic or a semi-direct product of cyclic groups. The expounded method of derivation of the line groups consists of first extending by the cyclic point groups and then obtaining the rest of the line groups by unification within the Euclidean group. To this purpose, a simple test is given to decide whether two line groups multiply into a third one. The method displays the subgroup structure of the line groups, relevant to the construction of their irreducible representations. All the line groups are derived and tabulated.