Similarity and Coincidence Isometries for Modules

TL;DR

This paper investigates the structure of similarity and coincidence isometries of modules in Euclidean space, revealing their algebraic composition and generalizing lattice results to quasicrystallography contexts.

Contribution

It characterizes the factor group of similarity modulo coincidence isometries for modules, extending lattice results to quasicrystallography applications.

Findings

Factor group is a direct sum of cyclic groups of prime power order

For prime dimension p, the factor group is an elementary Abelian p-group

Generalizes lattice isometry results to modules in quasicrystallography

Abstract

The groups of (linear) similarity and coincidence isometries of certain modules in d-dimensional Euclidean space, which naturally occur in quasicrystallography, are considered. It is shown that the structure of the factor group of similarity modulo coincidence isometries is the direct sum of cyclic groups of prime power orders that divide d. In particular, if the dimension d is a prime number p, the factor group is an elementary Abelian p-group. This generalizes previous results obtained for lattices to situations relevant in quasicrystallography.