Similarity versus Coincidence Rotations of Lattices

The groups of similarity and coincidence rotations of an arbitrary lattice L in d-dimensional Euclidean space are considered. It is shown that the group of similarity rotations contains the coincidence rotations as a normal subgroup. Furthermore, the structure of the corresponding factor group is examined. If the dimension d is a prime number, this factor group is an elementary Abelian d-group. Moreover, if L is a rational lattice, the factor group is trivial (d odd) or an elementary Abelian 2-group (d even).