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Show the group of units in Z_10 is a cyclic group of order 4

  1. May 17, 2015 #1
    1. The problem statement, all variables and given/known data

    Show that the group of units in Z_10 is a cyclic group of order 4
    2. Relevant equations


    3. The attempt at a solution
    group of units in Z_10 = {1,3,7,9}

    1 generates Z_4

    3^0=1, 3^1=3, 3^2=9, 3^3= 7, 3^4= 1, this shows <3> isomorphic with Z_4

    7^0=1 7^1= 7, 7^2= 9 7^3=3 7^4=1, this shows <7> isomorphic with Z_4

    9^0=1 9^1=9 9^2 =1 9^3=9 9^4=1, this shows <9> doesn't isomorphic with Z_4

    Did I do something wrong that I don't see this is a cyclic group of order 4?
     
  2. jcsd
  3. May 17, 2015 #2

    SammyS

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    You didn't expect 1 to generate the whole group either, did you?
     
  4. May 17, 2015 #3
    Right, 1 can't generate the whole. So there is only 3 and 7 isomorphic with Z_4.
     
  5. May 18, 2015 #4

    pasmith

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    You're done here: your group contains four elements, and you've shown that it contains an element of order 4. Therefore it is cyclic.
     
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