Similarity versus Coincidence Rotations of Lattices

TL;DR

This paper investigates the structure of similarity and coincidence rotation groups of lattices, revealing their relationships and properties, especially in prime dimensions and for rational lattices, with implications for understanding lattice symmetries.

Contribution

It characterizes the structure of the factor group of similarity rotations modulo coincidence rotations, including cases for prime dimensions and rational lattices.

Findings

Similarity rotations form a group containing coincidence rotations as a normal subgroup.

The factor group is an elementary Abelian d-group when d is prime.

For rational lattices, the factor group is trivial or an elementary Abelian 2-group depending on dimension.

Abstract

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).