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Subgroups generated by groups

  1. Nov 6, 2009 #1
    1. The problem statement, all variables and given/known data
    In the group <Z[tex]\stackrel{X}{13}[/tex]> of nonzero classes modulo 13 under multiplication, find the subgroup generated by [tex]\overline{3}[/tex] and [tex]\overline{10}[/tex]


    2. Relevant equations



    3. The attempt at a solution
    Doesnt 3 generate {3,6,9,12} and 10 generate {2,5,10}?
     
  2. jcsd
  3. Nov 6, 2009 #2

    HallsofIvy

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    The problem says says "under multiplication" so the subgroup generated by [itex]\overline{3}[/itex] includes all products of [itex]\overline{3}[/itex]. You are adding: 3+ 3= 6, etc. 3*3= 9 and 3*9= 27= 1 (mod 13)
     
  4. Nov 6, 2009 #3
    3*3 = 9 (13)
    3*3*3 =27 = 1 (13)
    3*3*3*3 = 81 = 3(13)
    3*3*3*3*3 = 243 = 9(13)
    3*3*3*3*3*3 = 1 (13)

    Do this, and then check that the results are allowed given the constraints you can infer from the order of the Cyclic Group.
     
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