We have already, in Section 9.5, met the notion of an abelian (or commutative) group with examples. Although this chapter is self-contained, the reader will find it helpful to read 9.5 again. The basic notion which we discuss in this chapter is that of a group, and an abelian group is then just a group in which the group operation is written as addition and is commutative (see G4 below). In fact, the notion of a group plays a fundamental role in algebra and geometry and an important role in the calculus and mathematical analysis. For example, the theory of groups virtually arose (in the hands of Galois, Abel, and others) out of the attempt to understand why equations of degree ≤4 could be solved by standard methods, whereas those methods failed for equations of degree ≥5. For the history of this successful attack on a very natural and classical problem the reader is referred to one of E. T. Bell’s books ( is probably the most attractive reference); for an account of the Galois theory of equations, the reader may consult Postnikov , the relevant chapter of Birkhoff-MacLane , or the more sophisticated text  by Artin.
KeywordsAbelian Group Group Operation Normal Subgroup Cyclic Group Quotient Group
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