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Classify the group Z4xZ2/0xZ2 using fund.thm. of finetely gen. abl. grps.

  1. Mar 22, 2012 #1

    Leb

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    1. The problem statement, all variables and given/known data

    Clasify the group Z4xZ2/{0}xZ2 using the fundamental theorem of finitely generated abelian groups.


    2. Relevant equations
    FTOFGAG: In short it states that every finitely generated abelian group G is isomorphic to a direct product of cyclic groups of the form Z(p1)r1x....xZ(pn)rnxZxZ...xZ

    where pn's are prime numbers and rn's are +ve integers (p's,r's can be same)

    Theorem 14.11 is just the fundamental theorem of homomorphism.

    3. The attempt at a solution

    As given in the book SolAB.jpg

    First time (at least from what I remember, and my memory span is that of a worm...) I encounter a projection map. But the main thing is, that I do not see how the theorem was applied ? I think I can justify to myself why pi(x,y) only gives x - otherwise, it would not be possible to make it isomorphic to Z4 (I mean it would not make any sense to map Z4xZ2 to only Z4), but why do we make such a choice in the first place ? And why can we say that {0}xZ2 is the kernel ?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 22, 2012 #2
    {0}xZ2 is the kernel of pi because pi(0xz2)=0 for any z2 in Z2. It can be shown that pi: Z4xZ2→Z4 is a homomorphism, therefore the canonical map Pi: Z4xZ2/{0}xZ2→Z4 is isomorphism, so Z4xZ2/{0}xZ2 and Z4 are isomorphic. I don't see why you need the theorem about finite abelian group, maybe this is just an example of that theorem.
     
  4. Mar 23, 2012 #3

    Leb

    User Avatar

    Thanks sunjin09!
    So it is all about how we define pi.
     
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