Presentation of Groups
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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.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