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Mutliplication table of quotient groups

  1. Nov 23, 2009 #1
    1. The problem statement, all variables and given/known data
    Write the multiplication table of C[tex]_{6}[/tex]/C[tex]_{3}[/tex]
    and identify it as a familiar group.


    2. Relevant equations



    3. The attempt at a solution
    C[tex]_{6}[/tex]={1,[tex]\omega[/tex],[tex]\omega^2[/tex],[tex]\omega^3[/tex],[tex]\omega^4[/tex],[tex]\omega^5[/tex]}
    C3={1,[tex]\omega[/tex],[tex]\omega^2[/tex]}
    The cosets are C3 and [tex]\omega^3[/tex]C3
    I just need help making the multiplication table.
     
  2. jcsd
  3. Nov 23, 2009 #2
    I'm assuming [itex]C_n[/itex] and [itex]C^n[/itex] both refer to the cyclic group of order n, since that's the impression I get from your post.

    if you meant for [itex]C_6[/itex] to be generated by [itex]\omega[/itex], then you should have [itex]C_3 = \{1,\omega^2,\omega^4\}[/itex] because otherwise [itex]C_3[/itex] is not a group. Then the cosets should be [itex]C_3[/itex], [itex]\omega C_3[/itex].

    What exactly are you having trouble with? As you said yourself the group [itex]C_6/C_3[/itex] has exactly two elements ([itex]C_3[/itex] and [itex]\omega C_3[/itex]), so the following four are the possible products you need to compute and insert in the multiplication table:
    [tex]C_3 \times C_3[/tex]
    [tex]C_3 \times \omega C_3[/tex]
    [tex]\omega C_3 \times C_3[/tex]
    [tex]\omega C_3 \times \omega C_3[/tex]
     
    Last edited: Nov 23, 2009
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