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Thread: Short exact sequences.....

  1. #1 Short exact sequences..... 
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    .....of groups. I am trying to study group extension theory, and find there is something I don't quite get. I wonder if anyone can help.

    Suppose is a group and . Then elementary group theory induces the quotient group , with as the identity, and a surjection , No problems here, I believe.

    The SES of homomorphisms is now defined by my text as

    which is OK too, since I am interpreting the "1" as a multiplicative unit (or possibly the identity matrix) so and are the obvious inclusion maps, is the surjection above and is the usual trivial homomorphism. Is this fair?

    I am now offered the following highly non-intuitive claims: from the above, we may conclude that, for any SES that this implies . I do not see the logic here. Any thoughts?

    More starling perhaps is the corollary for matrix (Lie) groups:

    and



    I am struggling to make sense of this; I can just about see that, as is a double covering for the first might, be true. But I can make no sense of the last. Any tips would be greatly appreciated.


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  3. #2 Re: Short exact sequences..... 
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    Quote Originally Posted by Guitarist
    .....of groups. I am trying to study group extension theory, and find there is something I don't quite get. I wonder if anyone can help.

    Suppose is a group and . Then elementary group theory induces the quotient group , with as the identity, and a surjection , No problems here, I believe.

    The SES of homomorphisms is now defined by my text as

    which is OK too, since I am interpreting the "1" as a multiplicative unit (or possibly the identity matrix) so and are the obvious inclusion maps, is the surjection above and is the usual trivial homomorphism. Is this fair?
    I don't know if it is fair, but I do know that it is correct.

    I am now offered the following highly non-intuitive claims: from the above, we may conclude that, for any SES that this implies . I do not see the logic here. Any thoughts?
    The key here is that the sequence is exact, i.e. the image of each arrow on the left is the kernel of the arrow on the right. So, H is the kernel of the homomorphism from G to A, and G is the kernel of the trivial map, so the map from G to A is surjective. From that you can conclude that G/H is isomorphic to A.

    More starling perhaps is the corollary for matrix (Lie) groups:

    and




    I am struggling to make sense of this; I can just about see that, as is a double covering for the first might, be true. But I can make no sense of the last. Any tips would be greatly appreciated.
    I assume that your problem is with the last isomorphism, since you say understand the first one.

    U(1) is the unitary group in one dimension, or the complex numbers of modulus 1, or the circle group. But the circle group is just the unit interval with the end points identified, taken as a group, which is R/Z.

    BTW which text are you using ?


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  4. #3  
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    Thanks for that. I am using a battered copy of Azcarraga & Izquierdo Lie, groups, Lie algebras, cohomology and some applications.... I found in a 2nd hand bookshop. I don't like it so well, as it uses some rather unfamiliar notation and terminology,

    Since you seem knowledgeable in this area, I may have some more questions, if that's OK. Stay tuned.
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