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Subgroups of Order Three

  1. May 23, 2008 #1
    [SOLVED] Subgroups of Order Three

    The problem statement, all variables and given/known data
    Let G be a subgroup containing exactly eight elements of order three. How many subgroups of order three does G have?

    The attempt at a solution
    This problem was discussed in class today. The professor said that G has four subgroups of order three. I didn't follow his explanation very well so I didn't understand why. Since there are eight elements of order three, wouldn't each of these elements constitute a subgroup of order three so G has at least eight subgroups of order three?
  2. jcsd
  3. May 23, 2008 #2
    Suppose [tex]a \in G[/tex] is order 3 then [tex]a^2[/tex] is also order 3. They belong to the same subgroup. That means that only 4 of the 8 are generators, and the other 4 are their squares.
  4. May 23, 2008 #3
    I see. I overlooked that fact. Thanks a lot.
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