Event Name

Sixth basic course on symmetry and group theory

Start Date 31st Jul 2017
End Date 4th Aug 2017

Aim and target

The training course is normally held biannually (March and August) and aims at providing a solid background in symmetry and group theory to students and scientists working on crystalline materials. It is divided in two branches:

  1. Basic course: open to everybody, no pre-requisite apart from basic matrix algebra:
  2. Advanced course: open to former participants of the basic course.

Both courses usually span five full days. Exercices proposed during the whole course are integral part of the course: participants are requested to actively take part in solving the proposed exercises. Evening sessions are proposed for discussion, practicals, solution to the exercises, questions & answers. Didactic material is distributed before and during the lectures and also made available in electronic form.


Basic course

  • Abstract algebra and the notion of group
  • Introduction to crystallographic symmetry and lattices
  • Morphological symmetry
  • Stereographic projection, with exercises
  • Crystallographic point groups and geometric crystal classes
  • Symmetry elements with a translational component (screw axes and glide planes)
  • Introduction to space groups: basic notions and the Hermann-Mauguin symbolism
  • Graphical construction of space groups
  • Practicals on the graphical construction of space groups
  • Metric tensor and applications
  • Matrix representation of symmetry operations
  • Change of axial settings and alternative settings of space groups
  • Normalizers of point and space groups
  • Chirality
  • Wyckoff positions, Wyckoff sets
  • Factor groups, symmorphic groups
  • Crystallographic orbits
  • Reciprocal lattice and systematic absences
  • Group-subgroup relations
  • Structure-substructure relations and applications to displacive phase transitions


The official language of the training course is Japanese. No simultaneous interpretation is provided.

Sokendai Interdisciplinary Lecture

From the fifth basic course, this training course has become part of the education provided by Sokendai, the Graduate University for Advanced Studies: University students can get credits through this university for attending the training course.

Lecture notes

The content of the lectures of the basic course is now partly being published in a series of didactic articles in the Journal of the Crystallographic Society of Japan. The first article is available from the J-Stage webiste at this address.


The training course is normally held at KEK, Tsukuba campus. Occasionally, alternative venues are considered, depending on the availability of the facilities and local organizers. The third basic was held at the Institute for Protein Research, Osaka University.


Prof. Massimo Nespolo, Universit√© de Lorraine, France (on behalf of MaThCryst)

Local Organizing Committee

Dr. Maki Okube, Tohoku University
Prof. Takashi Kamiyama, KEK
Prof. Noriyuki Igarashi, KEK

Further information

Details are available at the Japanese web site.

Location Tsukuba
Contact Yoshimi Takahashi
URL http://www.crystallography.fr/mathcryst/TrainingCourseJapan.php

Category Training courses
Topics Mathematical and Theoretical Crystallography