Unformatted text preview: W G ²! G= K . Then ı ' W G ²! G= K is a surjective homomorphism, because it is a com-position of surjective homomorphisms. The kernel of ı ' is the set of x 2 G such that '.x/ 2 ker . / D K ; that is, ker . ı '/ D ' ± 1 . K/ D K . According to the homomorphism theorem, Theorem 2.7.6 , G= K Š G= ker . ı '/ D G=K: More explicitly, the isomorphism G=K ²! G= K is xK 7! ı '.x/ D '.x/ K: Using the homomorphism theorem again, we can identify G with G=N . This identiﬁcation carries K to the image of K in G=N , namely K=N . Therefore, .G=N/=.K=N/ Š G= K Š G=K: n The following is a very useful generalization of the homomorphism theorem. We call it the Factorization Theorem, because it gives a condition...
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- Fall '08
- Algebra, Normal subgroup, Homomorphism, kernel, ı, Homomorphism theorem