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Question: Determine all the finite groups that have exactly one nontrivial proper subgroup.

MY attempt is that the order of group G has to be a positive nonprime integer n which has only one divisor since any divisor a of n will form a proper subgroup of order a. Since 4 is the only nonprime number that has only 1 divisor which is 2, All groups of order 4 has only 1 nontrivial proper subgroups (Z4 and D4)

  • 1
    $D_4$ is the Klein $4$-group, which has three non-trivial proper subgroups.2012-09-18
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    What about $9$?2012-09-18
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    Or 25? $ $ $ $ $ $2012-09-18
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    Your claim that any divisor of $n$ gives you a subgroup of that order is false: for example, $A_4$ has order $12$ but no subgroup of order $6$.2012-09-18
  • 1
    Hint: Pick an element outside your unique proper subgroup, and check the subgroup it generates...2012-09-19
  • 0
    http://math.stackexchange.com/q/154666/85812012-09-19

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