Skip to main content
SpringerLink
Log in

Weakly Associative Groups

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

The space ℝ 3 c of 3-dimensional relativistically admissible velocities possesses (i) a binary operation which represents the relativistic velocity composition law; and (ii) a mapping from the cartesian product ℝ 3 c ×ℝ 3 c into a subgroup of its automorphism group, Aut(ℝ 3 c ), representing the Thomas precession of special relativity. These binary operation and mapping are studied in special relativity as two isolated phenomena. It was recently discovered, however, that they are linked by an algebraic structure which gives rise to a theory of weakly associative and weakly associative-commutative groups. The axioms of these groups are presented in this paper and employed to obtain various interesting results. The algebraic structure underlying these nonstandard groups has been discovered and studied in a totally different context by Karzel (1965), Kerby and Wefelscheid.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M.A. Armstrong, Groups and Symmetry, Springer-Verlag, New York, 1988.

    Book  MATH  Google Scholar 

  2. W.A. Baylis and G. Jones, Special relativity with Clifford algebras and 2x2 matrices, and the exact product of two boosts,/. Math. Phys. 29 (1988), 57–62.

    Article  MathSciNet  MATH  Google Scholar 

  3. W. Benz, Vorlesungen über der Algebren, Springer-Verlag, New York 1973

    Book  MATH  Google Scholar 

  4. R.H. Brück, A Survey of Binary Systems, 2nd ed., Springer-Verlag, New York 1966.

    Google Scholar 

  5. H. Karzel, Inzidenzgruppen I, lecture notes by I. Pieper and K. Sörensen, Univ. Hamburg (1965), 123–135.

  6. H. Karzel, Zusammenhänge zwischen Fastbereichen, scharf 2-fach transitiven Permutationsgruppen und 2-Strukturen mit Rechtecksaxiom, Abh. Math. Sem. Univ. Hamburg 32 (1968), 191–206.

    Article  MathSciNet  MATH  Google Scholar 

  7. W. Kerby, Infinite Sharply Multiply Transitive Groups, Hamburger Mathematische Einelschriften, Neue Folge, Heft 6. Vandenhoek und Ruprecht, Göttingen 1974.

  8. W. Kerby and H. Wefelscheid, Bemerkungen über Fastbereiche und scharf 2-fach transitive Gruppen, Abh. Math. Sem. Univ. Hamburg 37 (1971), 20–29.

    Article  MathSciNet  Google Scholar 

  9. W. Kerby and H. Wefelscheid, Über eine scharf 3-fach transitiven Gruppen zugeordnete algebraische Struktur, Abh. Math. Sem. Univ. Hamburg 37 (1972), 225–235.

    Article  MathSciNet  MATH  Google Scholar 

  10. W. Kerby and H. Wefelscheid, Conditions of finiteness in sharply 2-transitive groups, Aequat. Math. 8(1972), 169–172.

    Article  MathSciNet  Google Scholar 

  11. W. Kerby and H. Wefelscheid, Über eine Klasse von scharf 3-fach transitiven Gruppen, J. reine angew Math. 268/269 (1974), 17–26.

    MathSciNet  Google Scholar 

  12. W. Kerby and H. Wefelscheid, The maximal subnear-field of a neardomain, J. Algebra, 28, (1974), 319–325.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Kist, Theorie der veralegemeinerten kinematischen Räume, Results Math. (Birkhäuser Verlag) 12 (1987), 325–347.

    Article  MathSciNet  MATH  Google Scholar 

  14. N.A. Salingaros, Erratum: The Lorentz group and the Thomas precession. II. Exact results for the product of two boosts, J. Math. Phys. 29 ( 1988), 1265.

    Article  MathSciNet  Google Scholar 

  15. I.N. Sneddon, ed., Encyclopedic Dictionary of Mathematics for Engineers and Applied Scientists, p. 320, Pergamon, New York, 1976.

  16. L.H. Thomas, The motion of the spinning electron, Nature 117 (1926), 514.

    Article  Google Scholar 

  17. L.H. Thomas, The kinematics of an electron with an axis, Philos. Mag. 3 (1927), 1–22.

    MATH  Google Scholar 

  18. L.H. Thomas, Recollections of the discovery of the Thomas precessional frequency, AIP Conf Proc. No. 95, High Energy Spin Physics Brookhaven National Lab, ed. G.M. Bunce, (1982), 4–12.

  19. G.E. Uhlenbeck, Fifty years of spin: personal reminiscences, Phys. Today, June (1976), 43–48.

  20. A.A. Ungar, Thomas rotation and the parametrization of the Lorentz transformation group, Found. Phys. Lett. 1 (1988), 57–89.

    Article  MathSciNet  Google Scholar 

  21. A.A. Ungar, The Thomas rotation formalism underlying a nonassociative group structure for relativistic velocities, Appl. Math. Lett. 1 (1988), 403–405.

    Article  MathSciNet  MATH  Google Scholar 

  22. A.A. Ungar, Axiomatic approach to the nonassociative group of relativistic velocities, Found. Phys. Lett. 2 (1989), 199–203.

    Article  MathSciNet  Google Scholar 

  23. A.A. Ungar, The relativistic velocity composition paradox and the Thomas rotation, Found. Phys. 19(1989), 1383–1394.

    Article  MathSciNet  Google Scholar 

  24. A.A. Ungar, Quasidirect product groups and the Lorentz transformation group, in T.M. Rassias (ed.), Constantin Caratheodory: An International Tribute, World Scientific Pub., NJ, 1991.

    Google Scholar 

  25. A.A. Ungar, The relativistic noncommutative nonassociative group of velocities and the Thomas rotation, Results Math. 16(1989), 168–179.

    Article  MathSciNet  MATH  Google Scholar 

  26. H. Wähling, Theorie der Fastkörper, Thaies Verlag, W. Germany, 1987.

    MATH  Google Scholar 

  27. H. Wefelscheid, ZT-Subgroups of sharply 3-transitive Groups, Proc. Edinburgh Math. Soc, 23, (1980), 9–14.

    Article  MathSciNet  MATH  Google Scholar 

  28. H. Wefelscheid, personal communication.

  29. H.E. Wolfe, Introduction to Non-Euclidean Geometry, p. vi, Dryden Press, New York, 1945.

    Google Scholar 

  30. H. Wussing, The Genesis of the Abstract Group Concept, p. 193(trans, by A. Shenitzer), MIT press, MA, 1984.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, North Dakota State University, Fargo, North Dakota, 58105, USA

    Abraham A. Ungar

Authors
  1. Abraham A. Ungar
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ungar, A.A. Weakly Associative Groups. Results. Math. 17, 149–168 (1990). https://doi.org/10.1007/BF03322638

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03322638

Keywords

Search

Navigation