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Up to isomorphism definition

  1. Nov 29, 2011 #1
    what does it really mean?

    for instance if asked to list all abelian grps of order 12 up to iso, then do we include Z12 or not?
     
  2. jcsd
  3. Nov 29, 2011 #2

    jgens

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    When you are asked to classify all groups of order 12 up to isomorphism, this means two things:
    • You need to create a list of groups such that every group of order 12 is isomorphic to one of these groups.
    • None of you the groups you listed are isomorphic to each other.
     
  4. Nov 30, 2011 #3

    micromass

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    Specifically: yes, you need to include [itex]\mathbb{Z}_{12}[/itex] or some group isomorphic to it. There are more groups of order 12 though...
     
  5. Nov 30, 2011 #4
    Yes thanks I get it now.

    Hence for ABELIAN groups of order 12 we have [itex]\mathbb{Z}_{12}[/itex] ≈ [itex]\mathbb{Z}_{3}[/itex] X [itex]\mathbb{Z}_{4}[/itex], and [itex]\mathbb{Z}_{2}[/itex] X [itex]\mathbb{Z}_{2} [/itex] X [itex]\mathbb{Z}_{3}[/itex]

    what would be [itex]\mathbb{Z}_{2}[/itex] X [itex]\mathbb{Z}_{6}[/itex] isomorphic to?
     
  6. Nov 30, 2011 #5

    micromass

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    In general: if gcd(a,b)=1, then [itex]\mathbb{Z}_{ab}\cong \mathbb{Z}_a\times \mathbb{Z}_b[/itex] (try to prove this!!).

    So we would have [itex]\mathbb{Z}_2\times \mathbb{Z}_6\cong \mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_3[/itex].
     
  7. Nov 30, 2011 #6
    we can construct an isomorphism ψ: from [itex]\mathbb{Z}_{ab}[/itex] to [itex]\mathbb{Z}_a\times \mathbb{Z}_b[/itex]

    such that ψ(x)= (x mod a, x mod b), it is a homomorphism, a surjection(for any y in the range, there exists an x congruent to z(mod ab) "the solution to the congruence system, and finally ψ is an injection because domain and codomain are finite sets with equal cardinality or order .
     
  8. Nov 30, 2011 #7

    micromass

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    Yes, but where did you use that gcd(a,b)=1?? You do need this!
     
  9. Nov 30, 2011 #8
    my computer just died.
    we used the fact that a and b are coprime to show that the system of equations has a solution by CRT.
     
  10. Nov 30, 2011 #9

    micromass

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    That seems alright!! :smile:
     
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