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# Similarity and Coincidence Isometries for Modules

Published:2011-04-25
Printed: Mar 2012
• Svenja Glied,
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany
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## Abstract

The groups of (linear) similarity and coincidence isometries of certain modules $\varGamma$ 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.
 MSC Classifications: 20H15 - Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25] 82D25 - Crystals {For crystallographic group theory, see 20H15} 52C23 - Quasicrystals, aperiodic tilings

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