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Cyclic group

  1. May 27, 2009 #1
    1. The problem statement, all variables and given/known data

    How do i go about proving that a group is cyclic?

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 27, 2009 #2

    dx

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    Start with the definition of a cyclic group, and see if your group satisfies it.
     
  4. May 27, 2009 #3
    The group, G, is a finte group with cardinality p, a prime integer. How should i start off, if i need to prove it's cyclic?
     
  5. May 27, 2009 #4

    dx

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    Are you familiar with Lagrange's theorem?
     
  6. May 27, 2009 #5
    yes, i know that the only subgroups of G are itself and the subgroup {e} which consists of the neutral element. This is because the only possibilities of the cardinalities of the subgroups are 1 or p.
     
  7. May 27, 2009 #6

    dx

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    Ok, now pick an element of the group G, say g not equal to 1. What are the possible orders of g?
     
  8. May 27, 2009 #7
    possible orders of g are 1 or p? Since those are the only numbers that divide the prime number p.
     
  9. May 27, 2009 #8

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    It cannot be 1 because we assumed g was not equal to the identity. So the order of g must be p, and therefore G = {1 , g, g2, ... , gp-1} which is cyclic.
     
  10. May 27, 2009 #9
    Ok, thanku very much for the help:)
     
  11. May 27, 2009 #10

    dx

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    No problem.
     
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