Maximal subgroup

A suitable way to represent group-subgroup relations is by means of family trees which show the relations from space groups to their maximal subgroups by arrows pointing downwards. In the middle of each arrow the kind of the relation and the index of the symmetry reduction are labeled, for example ... 

The symmetry reduction can be continued. A (non-maximal) subgroup of F43m is I42d with doubled lattice parameter c. On the way F43m —174 2d the Wyckoff position of the zinc atoms splits once more and can be occupied by atoms of two different elements. 

P63/m2/m2/c, which is the space group of the hexagonal closest-packing of spheres, has only one maximal subgroup in which the position 2a is split into two independent positions, namely P 3 2/m 1. If one of these positions is occupied and the other one remains vacant, this corresponds to the Cdl2 type. 

A case in point is the pentagonal subgroup Dsd of the icosahedral point group. This subgroup is a maximal subgroup, and the six pentagonal directions are equidistant , in the sense that any pair of them can be mapped onto any other pair. Induction then yields the five-fold degenerate H representation ... 

Note that the opposite is not necessarily true, e.g. the orbit of a maximal subgroup is not necessarily doubly transitive. A case in point in icosahedral symmetry is the trigonal subgroup. This is a maximal subgroup, but its orbit is not doubly transitive. In fact an icosahedron has ten trigonal sites which are however not all equidistant. Induction from D d yields three irreps ... 

The stabilizer of a vertex in a simplex, i.e. the group of all elements of S which leave a given vertex invariant, is the maximal subgroup 5 i. The set of all vertices thus will transform as the induced representation of a totally symmetric irrep of the site group in the parent group. Since this representation is certainly doubly... 

At present we have found that for the degenerate point group irreps which are listed in the table the basis functions can be expressed by means of a carrier space which exactly matches the orbit of a maximal subgroup of the point group, and counts G / H = n elements. The one-particle Hamiltonian operating in this carrier space can easily be constructed as follows ... 

So far the analysis has lead to the concept of a carrier space which links the degeneracy to a doubly transitive orbit of cosets of maximal subgroups. Interactions in this space are expressed as transition operators between the cosets. The final part of the treatment should bring in the vibrational degrees of freedom which are responsible for the Jahn-Teller activity. 

The usual JT effect in 3D point groups exemplifies the second case, Tyr C Tnm- These cases involve molecules that are more involved than the simplexes, which implies that the site which is stabilized by a maximal subgroup contains more nuclei than the one that is considered in the simplex. As a result all the possible JT interaction symmetries are represented at least by one normal mode, but in addition the space of vibrational modes also contains inactive modes. A case in point are centrosymmetric molecules where only gerade modes can be JT active, the odd modes are found in the remainder space Tnm — jt-... 

In this appendix we prove the corrolary to the theorem of Hall that the orbit of the cosets of a subgroup H of group G can only be doubly transitive for H a maximal subgroup of G. To this end we consider a further subgroup S c H, and examine if the orbit of cosets of S can be doubly transitive. Let gr and hp denote cosets representatives of H in G, and S in H resp., i.e. ... 

Clearly such an element cannot be found. Indeed from the first requirement it follows that gx must be in H, while the second condition places gx outside H. Hence only maximal subgroups can have doubly transitive cosets. 

MAXIMAL SUBGROUPS AND MINIMAL SUPERGROUPS OF A SPACE GROUP... 

SELECTION OF MAXIMAL SUBGROUPS OF SPACE GROUPS RELEVANT FOR THE EXAMPLES PRESENTED IN THE TEXT. 

The a 2 distortion is antisymmetric with respect to 3C2, a, and 2 3. As a result, when the mode is launched, all these symmetry elements will be destroyed, and the symmetry reduces to the subgroup C3 . In general, the result of a distortion will always be the maximal subgroup for which the distortion... 

---