Simple Groups

  • John StillwellEmail author
Part of the Undergraduate Texts in Mathematics book series (UTM)


We saw in Chapter 19 that the group concept came to light when Galois used it to explain why some equations are solvable and some are not. Solving an equation corresponds to “simplifying” a group by forming quotients, so knowing which equations are not solvable depends on knowing which groups cannot be “simplified.” These are the so-called simple groups. The groups associated with polynomial equations are finite, so one would like to classify the finite simple groups. Galois found one infinite family of such groups—the alternating groups An for n≥ 5—and three other provocative examples that we now view as the symmetry groups of finite projective lines. However, classification of the finite simple groups was much harder than could have been foreseen in the 19th century. It turned out to be easier (though still very hard) to classify continuous simple groups. This was done by Lie, Killing, and Cartan in the 1880s and 1890s. Each continuous simple group is the symmetry group of a space with hypercomplex coordinates, either from ℝ, ℂ,ℍ, or \(\mathbb{O}\). While this classification was in progress, it was noticed that a single continuous simple group can yield infinitely many finite simple groups, obtained by replacing the hypercomplex number system by a finite field. These “finite groups of Lie type” were completely worked out by 1960. Together with the alternating groups and the cyclic groups of prime order, they account for all but finitely many of the finite simple groups. But identifying all the exceptions—the 26 sporadic simple groups— turned out to be the hardest problem of all…


Simple Group Modular Function Continuous Group Finite Simple Group Unit Quaternion 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Akivis, M. A. and B. A. Rosenfeld (1993). Élie Cartan (1869–1951). Providence, RI: American Mathematical Society. Translated from the Russian manuscript by V. V. Goldberg.zbMATHGoogle Scholar
  2. Cartan, E. (1894). Sur la structure des groupes de transformations finis et continus. Paris: Nony et Co.Google Scholar
  3. Cartan, E. (1936). La topologie des espaces représentatives des groupes de Lie. L’Enseignement Math. 35, 177–200.Google Scholar
  4. Cole, F. N. (1893). Simple groups as far as order 660. Amer. J. Math. 15, 305–315.Google Scholar
  5. Coleman, A. J. (1989). The greatest mathematical paper of all time. Math. Intelligencer 11(3), 29–38.zbMATHCrossRefMathSciNetGoogle Scholar
  6. Feit, W. and J. G. Thompson (1963). Solvability of groups of odd order. Pacific J. Math. 13, 775–1029.zbMATHMathSciNetGoogle Scholar
  7. Freudenthal, H. (1951). Oktaven, Ausnahmegruppen und Oktavengeometrie. Mathematisch Instituut der Rijksuniversiteit te Utrecht, Utrecht.Google Scholar
  8. Gannon, T. (2006). Moonshine beyond the Monster. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press.zbMATHCrossRefGoogle Scholar
  9. Golay, M. (1949). Notes on digital encoding. Proc. IRE 37, 657.Google Scholar
  10. Griess, Jr., R. L. (1982). The friendly giant. Invent. Math. 69(1), 1–102.zbMATHCrossRefMathSciNetGoogle Scholar
  11. Hawkins, T. (2000). Emergence of the Theory of Lie Groups. New York: Springer-Verlag.zbMATHGoogle Scholar
  12. Killing, W. (1888). Die Zusammensetztung der stetigen endlichen Transformationsgruppen. Math. Ann. 31, 252–290.CrossRefMathSciNetGoogle Scholar
  13. Leech, J. (1967). Notes on sphere packings. Canad. J. Math. 19, 251–267.zbMATHMathSciNetGoogle Scholar
  14. Mathieu, E. (1861). Mémoire sur l’étude des fonctions des plusieurs quantités, sur le manière de les former et sur les substitutions qui les laissent invariables. J. Math. Pures Appl. 6, 241–323.Google Scholar
  15. McKean, H. and V. Moll (1997). Elliptic Curves. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  16. Moore, E. H. (1893). A doubly infinite system of simple groups. Bull. New York Math. Soc. 3, 73–78.CrossRefGoogle Scholar
  17. Paige, L. J. (1957). A note on the Mathieu groups. Canad. J. Math. 9, 15–18.zbMATHMathSciNetGoogle Scholar
  18. Ronan, M. (2006). Symmetry and the Monster. Oxford: Oxford University Press.zbMATHGoogle Scholar
  19. Stillwell, J. (2008). Naive Lie Theory. New York, NY: Springer-Verlag.zbMATHCrossRefGoogle Scholar
  20. Thompson, T. M. (1983). From Error-Correcting Codes through Sphere Packings to Simple Groups. Washington, DC: Mathematical Association of America.zbMATHGoogle Scholar
  21. Tits, J. (1956). Les groupes de Lie exceptionnels et leur interprétation géométrique. Bull. Soc. Math. Belg. 8, 48–81.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of San FranciscoSan FranciscoUSA

Personalised recommendations