Similar Submodules and Coincidence Site Modules

We consider connections between similar sublattices and coincidence site lattices and, more generally, between similar submodules and coincidence site modules of general (free) ℤ-modules in $ℝ^d$. In particular, we generalise results obtained by S. Glied and M. Baake on similarity and coincidence isometries of lattices and certain lattice-like modules called S-modules. An important result is that the factor group OS(M)/OC(M) is Abelian for arbitrary ℤ-modules M, where OS(M) and OC(M) are the groups of similar and coincidence isometries, respectively. In addition, we derive various relations between the indices of coincidence site lattices and their corresponding similar sublattices.