Magnetic line groups

Line groups can be used in symmetry considerations of quasi-one-dimensional systems (as well as their spin subsystems), stereoregular polymers, chain-type crystals, etc. Each line group is expressed in the form of a weak-direct product of two of its subgroups. One of them is an axial point group, describing the symmetry of the basic structural motive. The second factor is an infinite cyclic group (group of generalized translations), whose elements arrange the motives in the direction of the $z$ axis. With the use of these factorizations all of the 81 families of the magnetic line groups are found (any order of the principal axis is considered). The irreducible representations of the groups ${\mathit{L}n}_p$ are obtained in a form which yields selection rules suggesting conservation of a quasimomentum canonically conjugated to a helicoidal coordinate.