Presentation of Groups

  • Richard H. Crowell
  • Ralph H. Fox
Part of the Graduate Texts in Mathematics book series (GTM, volume 57)


In this chapter we give a firm foundation to the concept of defining a group by generators and relations. This is an important step; for example, if one is not careful to distinguish between the elements of a group and the words that describe these elements, utter confusion is likely to ensue.


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  1. 1.
    There are a number of similar problems which are known to have no general solution: deciding whether or not the group defined by a given presentation is trivial (the triviality problem), is finite, is abelian, is free, etc; deciding whether or not a given word is a consequence of a given set of words (the word problem); and many others. See M. O. Rabin, “Recursive Unsolvability of Group Theoretic Problems,” Annals of Mathematics, Vol. 67 (1958), pp. 172–194.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© R. H. Crowell and C. Fox 1963

Authors and Affiliations

  • Richard H. Crowell
    • 1
  • Ralph H. Fox
    • 2
  1. 1.Department of MathematicsDartmouth CollegeHanoverUSA
  2. 2.Princeton UniversityPrincetonUSA

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