Czechoslovak Journal of Physics B

, Volume 32, Issue 1, pp 3–18 | Cite as

An algebraic study of contemporary symmetry concepts — Colour groups and related symmetries

  • V. Kopský
Article
Received:

Abstract

After a few illustrative examples, an attempt is made to give an algebraic definition of symmetry, suitable for generalizations which are usually described as colour symmetry. The wreath product of groups Perm(A) and Perm(M) over the setM is shown to be the most general group of those bijections of cartesian productM×A which are compatible with the concept of colour point. The recently introduced concepts ofP-,Q-,Wp-, andWq-symmetries are discussed and it is shown thatQ-, andWq-symmetries can be, in a certain sense, reduced toP-, andWp-symmetries. The meaning of this reduction is briefly discussed for the case of spin groups.

Keywords

Colour General Group Wreath Product Related Symmetry Spin Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Flinders Petrie: Egyptian decorative art. London, 1920.Google Scholar
  2. [2]
    Speiser A.: Die Theorie der Gruppen von endlicher Ordnung. Berlin, 1937.Google Scholar
  3. [3]
    Heesch H.: Zs. Krist.71 (1929) 95.Google Scholar
  4. [4]
    Tavger B. A., Zaitsev V. M.: Zh. Eksp. & Teor. Fiz.30 (1956) 564; Sov. Phys. JETP3 (1956) 430.Google Scholar
  5. [4a]
    Opechowski W., Guccione R., Magnetism, Vol. IIA, ch. 3. (ed. Rado G. T. and Suhl H.). Academic Press, New York, 1965.Google Scholar
  6. [5]
    Shubnikov A. V.: Simmetriya i Antisymmetriya Konechnykh Figur. USSR Academy of Sciences, Moscow, 1951.Google Scholar
  7. [6]
    Belov N. V., Tarkhova T. N.: Sov. Phys. Cryst.1 (1956) 487.Google Scholar
  8. [7]
    Zamorzaiev A. M.: Teoriya prostoi i kratnoi simmetrii. Stiinca, Kishinev, 1976.Google Scholar
  9. [7a]
    Zamorzaiev A. M., Galiarskii E. I., Palistrant A. F.: Cvetnaia simmetriya, ee obobshcheniya i prilozheniya. Stiinca, Kishinev, 1978.Google Scholar
  10. [8]
    Niggli A., Wondratschek H.: Zs. Krist.114 (1960) 215.Google Scholar
  11. [8a]
    Wondratschek H., Niggli A.: Zs. Krist.115 (1961) 1.Google Scholar
  12. [9]
    Wittke O., Garrido J.: Bull. Soc. Frnc. Miner. Cryst.82 (1959) 223.Google Scholar
  13. [10]
    Litvin D. B., Opechowski W.: Physica76 (1974) 538.Google Scholar
  14. [10a]
    Litvin D. B.: Acta Cryst.A29 (1973) 651;A33 (1977) 279.Google Scholar
  15. [11]
    Koptsik V. A., Kotzev J. N.: Comm. Joint Inst. Nucl. Res. Dubna, USSR, E4-10788 (1977).Google Scholar
  16. [12]
    Koptsik V. A.: Kristall und Technik.10 (1975) 231.Google Scholar
  17. [13]
    Kotzev J. N.: Proc. Conf.: Kristallographische Gruppen. Bielefeld, 1979. match No 9 (1980), 41.Google Scholar
  18. [14]
    Dress A. W. M.: Proc. Conf.: Kristallographische Gruppen. Bielefeld, 1979. match No 9 (1980), 15.Google Scholar
  19. [15]
    Opechowski W.: Proc. Conf.: Kristallographische Gruppen Bielefeld, 1979, match No 9 (1980), 51.Google Scholar
  20. [16]
    Hall M.: The Theory of Groups. Macmillan, New York 1959.Google Scholar
  21. [17]
    Kurosh A. G.: Teoriya grupp. Moscow, Nauka, 1967.Google Scholar
  22. [18]
    Ascher E., Janner A.: Comm. Math. Phys.11 1968) 138.Google Scholar
  23. [19]
    Wigner E.: Group Theory and its Application to Quantum Mechanics of Atomic Spectra. New York, 1959.Google Scholar

Copyright information

© Academia, Publishing House of the Czechoslovak Academy of Sciences 1982

Authors and Affiliations

  • V. Kopský
    • 1
  1. 1.Institute of PhysicsCzechosl. Acad. Sci.Praha 8Czechoslovakia

Personalised recommendations