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Short presentations for crystallographic groups.

Igor A Baburin1

  • 1Ludwig-Maximilians-Universität München, Sektion Kristallographie, Theresienstrasse 41, 80333 München, Germany.

Acta Crystallographica. Section A, Foundations and Advances
|December 19, 2025
PubMed
Summary
This summary is machine-generated.

This study presents a practical method for creating concise group presentations for Euclidean crystallographic groups. Short presentations are linked to cycles in Cayley graphs, offering insights into group structure.

Keywords:
Cayley graphscrystallographic groupsfinitely presented groupsperiodic graphsstrong rings

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Area of Science:

  • Group Theory
  • Crystallography
  • Computational Mathematics

Background:

  • Group presentations are fundamental in abstract algebra.
  • Euclidean crystallographic groups are essential in understanding crystal structures.
  • Constructing efficient group presentations is computationally challenging.

Purpose of the Study:

  • To develop a practical approach for generating short presentations of Euclidean crystallographic groups.
  • To establish a connection between group relators and cycles in Cayley graphs.
  • To compute presentations for high-symmetry periodic graphs.

Main Methods:

  • Defining 'short presentation' based on the number and length of relators.
  • Analyzing the relationship between relators and cycles in Cayley graphs.
  • Utilizing the concept of 'strong rings' in Cayley graphs.
  • Computing presentations for specific classes of vertex-transitive groups.

Main Results:

  • A short presentation typically corresponds to strong rings in the Cayley graph.
  • This correspondence provides a natural upper bound for the size of relators.
  • Presentations were successfully computed for 2-, 3-, and 4-periodic graphs.
  • Connections between graph geodesics and quotient cycles were explored.

Conclusions:

  • The proposed method offers an efficient way to obtain short group presentations.
  • Cayley graph structures, particularly strong rings, are key to constructing concise presentations.
  • The findings are applicable to various dimensionalities and periodicities of crystallographic groups.