Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 009, 18 pages      arXiv:1101.3055      https://doi.org/10.3842/SIGMA.2011.009
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Introduction to Sporadic Groups

Luis J. Boya
Departamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain

Received September 18, 2010, in final form January 12, 2011; Published online January 16, 2011

Abstract
This is an introduction to finite simple groups, in particular sporadic groups, intended for physicists. After a short review of group theory, we enumerate the 1+1+16=18 families of finite simple groups, as an introduction to the sporadic groups. These are described next, in three levels of increasing complexity, plus the six isolated ''pariah'' groups. The (old) five Mathieu groups make up the first, smallest order level. The seven groups related to the Leech lattice, including the three Conway groups, constitute the second level. The third and highest level contains the Monster group M, plus seven other related groups. Next a brief mention is made of the remaining six pariah groups, thus completing the 5+7+8+6=26 sporadic groups. The review ends up with a brief discussion of a few of physical applications of finite groups in physics, including a couple of recent examples which use sporadic groups.

Key words: group theory; finite groups.

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References

  1. Bonolis L., From the rise of the group concept to the stormy onset of group theory in the new quantum mechanics. A saga of the invariant characterization of physical objects, events and theories, Rivista del Nuovo Cimento 027 (2004), 1-110.
  2. Weyl H., Theory of groups and quantum mechanics, Dover, New York, 1928.
  3. Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A., Atlas of finite simple groups. Maximal subgroups and ordinary characters for simple groups, Oxford University Press, Eynsham, 1985.
  4. Thomas A.D., Wood G.V., Group tables, Shiva Mathematics Series, Vol. 2, Shiva Publishing Ltd., Cambridge, Mass., 1980.
  5. Ramond P., Group theory. A physicist's survey, Cambridge University Press, Cambridge, 2010.
  6. Hall M. Jr., The theory of groups, The Macmillan Co., New York, 1959.
  7. Wigner E.P., Group theory and its application to the quantum mechanics of atomic spectra, Pure and Applied Physics, Vol. 5, Academic Press, New York - London, 1959.
  8. Robinson D.J.S., A course in the theory of groups, 2nd ed., Graduate Texts in Mathematics, Vol. 80, Springer-Verlag, New York, 1996.
  9. Bogopolski O., Introduction to group theory, European Mathematical Society (EMS), Zürich, 2008.
  10. Coxeter H.S.M., Regular polytopes, Dover Publications, Inc., New York, 1973.
  11. Conway J.H., Smith D.A., On quaternions and octonions: their geometry, arithmetic, and symmetry, A K Peters, Ltd., Natick, MA, 2003.
  12. Besche H.U., Eick B., O'Brien E.A., A millennium project: constructing small groups, Internat. J. Algebra Comput. 12 (2002), 623-644.
  13. Coxeter H.S.M., Moser W.O.J., Generators and relations for discrete groups, Springer-Verlag, New York - Heidelberg, 1972.
  14. Carter R.W., Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London - New York - Sydney, 1972.
  15. Artin E., Geometric algebra, Interscience Publishers, Inc., New York - London, 1957.
  16. Dieudonné J., La géométrie des groupes classiques, Springer-Verlag, Berlin - Göttingen - Heidelberg, 1955.
  17. Boya L.J., Campoamor-Stursberg R., Composition algebras and the two faces of G2, Int. J. Geom. Methods Mod. Phys. 7 (2010), 367-378, arXiv:0911.3387.
  18. Conway J.H., Sloane N.J.A., Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften, Vol. 290, Springer-Verlag, New York, 1988.
  19. Griess R.L. Jr., Twelve sporadic groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.
  20. Biggs N.L., White A.T., Permutation groups and combinatorial structures, London Mathematical Society Lecture Note Series, Vol. 33, Cambridge University Press, Cambridge - New York, 1979.
  21. Boya L.J., New derivation of Mathieu groups, to appear.
  22. Greenberg P., Mathieu groups, Courant Institute of Mathematical Sciences, New York University, New York, 1973.
  23. Coxeter H.S.M., The beauty of geometry. Twelve essays, Dover Publications, Inc., Mineola, NY, 1999.
  24. Eguchi T., Ooguri H., Tachikawa Y., Notes on the K3 Surface and the Mathieu group M24, arXiv:1004.0956.
  25. Ganon T., Monstrous moonshine: the first twenty-five years, Bull. London Math. Soc. 38 (2006), 1-33.
  26. Ronan M., Symmetry and the Monster. One of the greatest quests of mathematics, Oxford University Press, Oxford, 2006.
  27. Frenkel I., Lepowsky J., Meurman A., Vertex operator algebras and the Monster, Pure and Applied Mathematics, Vol. 134, Academic Press, Inc., Boston, MA, 1988.
  28. Harada K., "Moonshine" of finite groups, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2010.
  29. Ishimori H., Kobayashi T., Ohki H., Okada H., Shimizu Y., Tanimoto M., Non-Abelian discrete symmetries in particle physics, Prog. Theor. Phys. (2010), suppl. 183, 1-163, arXiv:1003.3552.
  30. Borcherds R.E., Sporadic groups and string theory, in First European Congress of Mathematics, Vol. I (Paris, 1992), Progr. Math., Vol. 119, Birkhäuser, Basel, 1994, 411-421.
  31. Borcherds R.E., Book review: "Moonshine beyond the Monster. The bridge connecting algebra, modular forms and physics" by T. Gannon, Bull. Amer. Math. Soc. 45 (2008), 675-679.
  32. Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
  33. Witten E., Three-dimensional gravity revisited, arXiv:0706.3359.
  34. Cheng M.C.N., K3 surface, N=4 dyons, and the Mathieu group M24, arXiv:1005.5415.

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