Examples of Groups: Infinite Discrete Groups

  • Igor R. Shafarevich
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 11)


We proceed now to consider infinite groups. Of course, the purely negative characteristic of not being finite does not reflect the situations which really arise. Usually the infinite set of elements of a group is defined by some constructive process or formula. This formula may contain some parameters, which may take integer values, or be real numbers, or even points of a manifold. This is the starting point of an informal classification: groups are called discrete in the first case, and continuous in the second. The simplest example of a discrete group is the infinite cyclic group, whose elements are of the form g n where n runs through all the integers.


Riemann Surface Symmetry Group Normal Subgroup Meromorphic Function Fundamental Group 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Igor R. Shafarevich
    • 1
  1. 1.Steklov Mathematical InstituteRussian Academy of ScienceMoscowRussia

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