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Show that a group with no proper nontrivial subgroups is cyc

  1. Feb 20, 2017 #1
    1. The problem statement, all variables and given/known data
    Show that a group with no proper nontrivial subgroups is cyclic.

    2. Relevant equations


    3. The attempt at a solution
    If a group G has no proper nontrivial subgroups, then its only subgroups are ##\{e \}## and ##G##. Assume that G has at least two elements, and let ##a## be any element besides ##e##. Then ##a## generates a subgroup of ##G##, but ##G## has no proper nontrivial subgroups, which means that ##a## must generate ##G##, so ##G## is cyclic.

    I feel that I am on the right track, but I also don't feel like I am being rigorous enough.
     
  2. jcsd
  3. Feb 21, 2017 #2

    andrewkirk

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    Your proof is fine. I would just replace "##a## generates a subgroup of ##G##" by "##a## generates a non-trivial subgroup of ##G##", and observe that ##\langle a\rangle##, the subgroup generated by ##a##, cannot be proper, before stating that ##\langle a\rangle=G##.
     
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