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Creating a homomorphism with given generators

  1. Apr 18, 2007 #1
    I got the Dihedral Group D = <(1 2 3 4 5 6 ), (1 2)(3 4)(5 6)> and the symmetric group Sym(5).

    Now I want to construct a homomorphism f : D --> Sym(5). Am I free to map the generators (1 2 3 4 5 6) and (1 2)(3 4)(5 6) to any element in Sym(5) as long holds:
    f((1 2 3 4 5 6))6 = 1,
    f((1 2)(3 4)(5 6))2 = 1.


    I tried
    f((1 2 3 4 5 6)) = (1 2 3)(4 5),
    f((1 2)(3 4)(5 6)) = (1 2).

    Which seems to be fine but
    f((1 2 3 4 5 6)) = (1 2 3)(4 5),
    f((1 2)(3 4)(5 6)) = (1 4).

    seems to fail?

    Why?
     
  2. jcsd
  3. Apr 18, 2007 #2

    matt grime

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    You can't just send them to *any* elements of the right order. Homomrphisms must preserve group structure. Like composition of elements.
     
  4. Apr 18, 2007 #3
    True but how do i know my first attempt is indeed a homomorphism and the 2nd one not. Now I "prooved" it by a computer program (MAGMA).
     
  5. Apr 18, 2007 #4

    matt grime

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    You check if it is a homomorphism. You know what the definition of a homomorphism is, so check if the maps satisfy the definition.
     
  6. Apr 18, 2007 #5

    matt grime

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    And remember that the dihedral group is defined by the relations, in this case, g^6=e, h^2=e and hgh=g^{-1}.
     
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