The lecture starts with a reminder of some basic facts and results on group-subgroup relations. It is followed by an exercise block on subgroups of crystallographic point groups, coset decomposition of group with respect to a subgroup, conjugate and normal subgroups, factor groups.

The second part of the lecture is focused on group-subgroup relations between space groups, including: different types of subgroups of space groups as t-subgroups, k-subgroups and general subgroups; symmetry relations between Wyckoff positions for a group-subgroup pair, supergroups of space groups. The normalizers of space groups are introduced and their importance in the determination of equivalent structure descriptions is discussed. The group-subgroup relations between space groups are applied in the analysis of possible relationships between crystal structures and the construction of the so-called Baernighausen trees.

The main points of the lecture:

1. Group-subgroup relations (general considerations): index, coset decomposition and normal subgroups; conjugate elements and conjugate subgroups; factor groups and homomorphism.
2. Group-subgroup relations between space groups:
   1. Subgroups of space groups: types of subgroups of space groups; Hermann theorem; maximal subgroups; series of isomorphic subgroups.
   2. Coset decomposition for a group-subgroup pair of space groups.
   3. Chains of maximal subgroups for a general group-subgroup pair. Contracted and complete graphs of maximal subgroups for a group-subgroup pair.
   4. Relations of Wyckoff positions for a group-subgroup pair.
3. Supergroups of space groups.
4. Normalizers of space groups; Affine and Euclidean normalizers; Wyckoff sets; Euclidean normalizers for specialized metrics.
5. Crystal-structure descriptions II.
   1. Equivalent crystal structure descriptions; structure descriptions compatible with reduced (subgroup) symmetry.
   2. Relations between crystal structures; Baernighausen trees constructions.

 

Supplementary Material