Unformatted text preview: G=N follows from repeated use of the deﬁnition 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 deﬁning the product in G=N is developed in Exercise 2.7.2 ....
View
Full Document
 Fall '08
 EVERAGE
 Algebra, Group Theory, Sets, Normal subgroup, Coset, quotient group construction, G=N

Click to edit the document details