Abstract: "The symmetries of infinite polymer chain molecules are discussed using the concept of line groups. Explicit directions are given for obtaining the irreducible representations of line groups employing the formalism developed for three-dimensional space groups. Line groups containing primitive and nonprimitive operations are discussed. Detailed considerations of the line group Vh are given with specific applications to the polyethylene molecule. Directions for generating symmetry-adapted vibrational coordinates for chain molecules are outlined."