Group Theory

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


The next three chapters are concerned with the emergence of “modern,” or abstract, algebra from the old algebra of equations. In the present chapter we look at group theory. Group theory today is often described as the theory of symmetry, and indeed groups have been inherent in symmetric objects since ancient times. However, extracting algebra from a symmetric object is a highly abstract exercise, and groups first appeared in situations where some algebra was already present. One of the first nontrivial examples was the group of integers mod p, for prime p, used by Euler (1758) to prove Fermat’s little theorem. Of course, Euler had no idea that he was using a group. But he did use one of the characteristic group properties, namely, the existence of inverses. Likewise, Lagrange (1771) was not aware of the group concept when he studied permutations of the roots of equations. But he was using the group S n of permutations of n things, and some of its subgroups. It was Galois (1831a) who first truly grasped the group concept, and he used it brilliantly to explain what makes an equation solvable by radicals. In particular, he was able to explain why the general quintic equation is not solvable by radicals. These discoveries changed the face of algebra, though few mathematicians realized it at first. In the second half of the 19th century the group concept spread from algebra to geometry, following the observation of Klein (1872) that each geometry is characterized by a group of transformations. This very fruitful idea is explored further in Chapter 23.


Normal Subgroup Simple Group Projective Transformation Linear Fractional Transformation Finite Simple Group 


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  1. Dickson, L. E. (1903). Introduction to the Theory of Algebraic Equations. New York: Wiley.zbMATHGoogle Scholar
  2. Dyck, W. (1882). Gruppentheoretische Studien. Math. Ann. 20, 1–44.CrossRefMathSciNetGoogle Scholar
  3. Dyck, W. (1883). Gruppentheoretische Studien II. Math. Ann. 22, 70–108.CrossRefMathSciNetGoogle Scholar
  4. Edwards, H. M. (1984). Galois Theory. New York: Springer-Verlag.zbMATHGoogle Scholar
  5. Gray, J. (1982). From the history of a simple group. Math. Intelligencer 4(2), 59–67.zbMATHCrossRefMathSciNetGoogle Scholar
  6. Magnus, W. (1930). Über diskontinuierliche Gruppen mit einer definierenden Relation (der Freiheitssatz). J. reine und angew. Math. 163, 141–165.zbMATHGoogle Scholar
  7. Rothman, T. (1982). Genius and biographers: the fictionalization of Évariste Galois. Amer. Math. Monthly 89(2), 84–106.zbMATHCrossRefMathSciNetGoogle Scholar
  8. Wussing, H. (1984). The Genesis of the Abstract Group Concept. Cambridge, MA.: MIT Press. Translated from the German by Abe Shenitzer.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of San FranciscoSan FranciscoUSA

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