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Lattice diagrams and generator in Algebra

  1. Jun 7, 2009 #1
    1. The problem statement, all variables and given/known data

    I do not understand the following statement (Please, see the attachment):

    "C4 has trivial subgroups and only one cyclic subgroup of 2 elements, namely <b>. This is because both a and c can be verified to be generators of C4."

    3. The attempt at a solution

    The notation <something> is normally used to indicate a generator of a group.
    However, the paragraph uses the notation only for the cyclic subgroup b such that <b>.

    The following support statement
    for the above clause is what I do not understand:
    "This is because both a and c can be verified to be generators of C4."

    If a subgroup has a generator, then it is a cyclic group.
    The paragraph says that the group has two generators, a and c.
    Then, a and c must be also cyclic subgroups of C4.

    This is a contradiction to the first clause that C4 has only one generator b.

    What does the paragraph really mean?
     

    Attached Files:

  2. jcsd
  3. Jun 7, 2009 #2
    I think they mean is that you can choose either a or c as the generator of the group. You only need one generator, and both elements are candidates for it.
     
  4. Jun 7, 2009 #3
    Thank you for your answer!

    Do you mean that a and c are subgroups of b?
    It seems that if a subgroup has two elements, then these two elements are subgroups too.

    If they are subgroups of b and b is a generator, then a and c seems to get the "generator" property from b.
     
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