Symmetry restrictions on phase transitions imposed by group-subgroup structure

The ideas of little groups and orbits of a group action as reviewed by Michel are used for a reexamination of Birman's subduction criterion and Ascher's discussion of the inverse Landau problem. As an illustration of the ease of application of these concepts from differential geometry we present a classification of the subgroups of $O_h^5\left(Fm3m\right)$ that result from distortions with symmetries of a single wave vector from one of the irreducible representations at the $\Gamma$, $X$, $L$, or $W$ point. These are the representations that are important in commensurate structural phase transitions.