Results in Mathematics

, Volume 17, Issue 1–2, pp 149–168 | Cite as

Weakly Associative Groups

  • Abraham A. Ungar


The space c 3 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 c 3 × c 3 into a subgroup of its automorphism group, Aut( c 3 ), 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.


Special Relativity Binary Operation Semidirect Product Identity Automorphism Thomas Precession 
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  1. [1]
    M.A. Armstrong, Groups and Symmetry, Springer-Verlag, New York, 1988.MATHCrossRefGoogle Scholar
  2. [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.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    W. Benz, Vorlesungen über der Algebren, Springer-Verlag, New York 1973MATHCrossRefGoogle Scholar
  4. [4]
    R.H. Brück, A Survey of Binary Systems, 2nd ed., Springer-Verlag, New York 1966.Google Scholar
  5. [5]
    H. Karzel, Inzidenzgruppen I, lecture notes by I. Pieper and K. Sörensen, Univ. Hamburg (1965), 123–135.Google Scholar
  6. [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.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    W. Kerby, Infinite Sharply Multiply Transitive Groups, Hamburger Mathematische Einelschriften, Neue Folge, Heft 6. Vandenhoek und Ruprecht, Göttingen 1974.Google Scholar
  8. [8]
    W. Kerby and H. Wefelscheid, Bemerkungen über Fastbereiche und scharf 2-fach transitive Gruppen, Abh. Math. Sem. Univ. Hamburg 37 (1971), 20–29.MathSciNetCrossRefGoogle Scholar
  9. [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.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    W. Kerby and H. Wefelscheid, Conditions of finiteness in sharply 2-transitive groups, Aequat. Math. 8(1972), 169–172.MathSciNetCrossRefGoogle Scholar
  11. [11]
    W. Kerby and H. Wefelscheid, Über eine Klasse von scharf 3-fach transitiven Gruppen, J. reine angew Math. 268/269 (1974), 17–26.MathSciNetGoogle Scholar
  12. [12]
    W. Kerby and H. Wefelscheid, The maximal subnear-field of a neardomain, J. Algebra, 28, (1974), 319–325.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    G. Kist, Theorie der veralegemeinerten kinematischen Räume, Results Math. (Birkhäuser Verlag) 12 (1987), 325–347.MathSciNetMATHCrossRefGoogle Scholar
  14. [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.MathSciNetCrossRefGoogle Scholar
  15. [15]
    I.N. Sneddon, ed., Encyclopedic Dictionary of Mathematics for Engineers and Applied Scientists, p. 320, Pergamon, New York, 1976.Google Scholar
  16. [16]
    L.H. Thomas, The motion of the spinning electron, Nature 117 (1926), 514.CrossRefGoogle Scholar
  17. [17]
    L.H. Thomas, The kinematics of an electron with an axis, Philos. Mag. 3 (1927), 1–22.MATHGoogle Scholar
  18. [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.Google Scholar
  19. [19]
    G.E. Uhlenbeck, Fifty years of spin: personal reminiscences, Phys. Today, June (1976), 43–48.Google Scholar
  20. [20]
    A.A. Ungar, Thomas rotation and the parametrization of the Lorentz transformation group, Found. Phys. Lett. 1 (1988), 57–89.MathSciNetCrossRefGoogle Scholar
  21. [21]
    A.A. Ungar, The Thomas rotation formalism underlying a nonassociative group structure for relativistic velocities, Appl. Math. Lett. 1 (1988), 403–405.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    A.A. Ungar, Axiomatic approach to the nonassociative group of relativistic velocities, Found. Phys. Lett. 2 (1989), 199–203.MathSciNetCrossRefGoogle Scholar
  23. [23]
    A.A. Ungar, The relativistic velocity composition paradox and the Thomas rotation, Found. Phys. 19(1989), 1383–1394.MathSciNetCrossRefGoogle Scholar
  24. [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. [25]
    A.A. Ungar, The relativistic noncommutative nonassociative group of velocities and the Thomas rotation, Results Math. 16(1989), 168–179.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    H. Wähling, Theorie der Fastkörper, Thaies Verlag, W. Germany, 1987.MATHGoogle Scholar
  27. [27]
    H. Wefelscheid, ZT-Subgroups of sharply 3-transitive Groups, Proc. Edinburgh Math. Soc, 23, (1980), 9–14.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    H. Wefelscheid, personal communication.Google Scholar
  29. [29]
    H.E. Wolfe, Introduction to Non-Euclidean Geometry, p. vi, Dryden Press, New York, 1945.Google Scholar
  30. [30]
    H. Wussing, The Genesis of the Abstract Group Concept, p. 193(trans, by A. Shenitzer), MIT press, MA, 1984.MATHGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 1990

Authors and Affiliations

  • Abraham A. Ungar
    • 1
  1. 1.Department of MathematicsNorth Dakota State UniversityFargoUSA

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