It's all in the group

The space group of a crystal structure is usually given by augmented matrices representing the action as affine mappings on direct space, but can also be described by generators and defining relators, i.e. by a group presentation. Related to the latter, the Cayley graph of a group is constructed in which the vertices correspond to the group elements and two vertices are connected by an edge if one is the product of the other with one of the generators. Baburin [Acta Cryst. (2026), A82, 18-31] shows how combinatorial and geometric information about a crystal structure and its symmetry group can be derived from the interplay between the Cayley graph and the group presentation.