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Unformatted text preview: W G ! G= K . Then ' W G ! G= K is a surjective homomorphism, because it is a com-position of surjective homomorphisms. The kernel of ' is the set of x 2 G such that '.x/ 2 ker . / D K ; that is, ker . '/ D ' 1 . K/ D K . According to the homomorphism theorem, Theorem 2.7.6 , G= K G= ker . '/ D G=K: More explicitly, the isomorphism G=K ! G= K is xK 7! '.x/ D '.x/ K: Using the homomorphism theorem again, we can identify G with G=N . This identication carries K to the image of K in G=N , namely K=N . Therefore, .G=N/=.K=N/ G= K 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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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