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Unformatted text preview: G=N follows from repeated use of the denition of the product, and the associativity of the product on G ; namely .aNbN/cN D abNcN D .ab/cN D a.bc/N D aNbcN D aN.bNcN/: It is clear that N itself serves as the identity for this multiplication and that a 1 N is the inverse of aN . Thus G=N with this multiplication is a group. Furthermore, is a homomorphism because .ab/ D abN D aNbN D .a/.b/: The uniqueness of the product follows simply from the surjectivity of : in order for to be a homomorphism, it is necessary that aNbN D abN . n The group G=N is called the quotient group of G by N . The map W G ! G=N is called the quotient homomorphism. Another approach to dening the product in G=N is developed in Exercise 2.7.2 ....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Sets

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