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Groups and kernels?

  1. Jan 13, 2008 #1
    1. The problem statement, all variables and given/known data
    Expain why every normal subgroup is the kernel of some homomorphism.

    3. The attempt at a solution
    Every kernel is a normal subgroup but the reverse I can't show rigorously. It seems possible how to show?
  2. jcsd
  3. Jan 13, 2008 #2


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    What's the obvious map from G to G/N? What's its kernel?
  4. Jan 13, 2008 #3


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    I would have thought that would be easy- it's the direction emphasised in Algebra texts! Of course, this says "some" homomorphism- you have to pick the homomorphism carefully.

    If H is a normal subgroup G, then we can define the "quotient group", G/H. There is an obvious homomorphism from G to H. What is the kernel of that homomorphism?

    Darn, I had to stop in the middle to take a telephone call and morphism got in in front of me!
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