Subgroups of weak-direct products and magnetic axial point groups

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Abstract
It is shown that all halving subgroups of a group G which is a weak-direct product of two of its subgroups H and K, can be constructed using halving subgroups of H and K. Similarly, if K is of order two, one can find all subgroups of G via the subgroups of H. Using the former method, all of the 31 families of magnetic axial point groups of arbitrary order are determined. These groups are of interest when ferromagnetic and ferroelectric phases of quasi-one-dimensional systems are considered. Also, it is demonstrated that those among the non-crystallographic magnetic axial point groups which are compatible with ferromagnetism (ferroelectricity), admit magnetisation (polarisation) only along the principle axis of rotation.

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