Theorem II.1 shows that every subgroup M of Z is either 0 or generated by its smallest element m>0; in the latter case it is generated by m or also by —m, but by no other element of M. For cyclic groups, we have:
  • Theorem VIII.1. Let G be a cyclic group of order m, generated by an element x. Let G’ be a subgroup of G; then there is a divisor d of m such that G’ is the cyclic group of order \( \frac{m}{d} \) generated by x d .


Cyclic Group Multiplicative Group Congruence Relation Congruence Class Decimal Fraction 
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.


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Copyright information

© Springer-Verlag New York Inc. 1979

Authors and Affiliations

  • André Weil
    • 1
  1. 1.Institute for Advanced StudyPrincetonUSA

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