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Math Help - Well-Defined function

  1. #1
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    Well-Defined function

    I have a more general question about this, but thought I'd use a specific example instead.
    I was trying to prove the Second Iso. Theorem for groups. Say G is a group and N a normal subgroup. I define the function f from {subgroups of G/N} to {subgroups of G containing N} by
     f(H') = \{g \in G : gN \in H' \} .
    I know this is well-defined but I would like to prove it. I know in general you take two elements in the domain, suppose they are equal and try to derive their image is equal. Then let
     \{ h_{1}N, h_{2}N,... \} = \{k_{1}N, k_{2}N,... \} .
    Then you can identify cosets on the left with cosets on the right, so  h_{i}N=k_{j}N . This only tells me that  k_{j}^{-1}h_{i} is in N. What do I do with this information? Should I do it some other way?
    Am I over-estimating the fact its well defined-ness is not obvious? I know in general the question of well-defined comes up if, for example in equivalence classes, there is ambiguity of representative. Here there doesn't really seem to be such a problem, right? I don't know, I recently started to realise I should question whether functions I construct are well-defined or not, but I don't know when it's necessary to so or when it's obvious it is.
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  2. #2
    Senior Member roninpro's Avatar
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    The map you have defined does not seem to have anything to do with the Second Isomorphism Theorem that I know. Are you trying to prove this? Lattice theorem - Wikipedia, the free encyclopedia

    If you think about your map for a moment, you realise that it simply regurgitates the contents of all of the cosets in H'; therefore, the map does not depend on representatives at all.

    I also think that your map is just a complicated way of defining \pi^{-1}(H'), where \pi:G\to G/N is the canonical homomorphism.
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  3. #3
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    Yeah it is the correspondence theorem. Sorry, I've always been introduced to the theorem (both for rings and groups) as the Second Isomorphism Theorem. Even the book I'm using does so... I don't really know why some people label it this way.

    I also think that your map is just a complicated way of defining , where is the canonical homomorphism.
    Well the way I was going about it is define two functions, one in one direction and one in the other. To show the required correspondence, show they are inverse to each other.
    The reason I did it this way is because I was having trouble proving injectivity with  \pi . But that's another problem.
    Thank you for the help!
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