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Math Help - Prove

  1. #1
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    Prove

    Suppose that G is a group that has exactly one nontrivial proper subgroup. Prove that G is cyclic and |G|=p^2 where p is prime.
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    Quote Originally Posted by mandy123 View Post
    Suppose that G is a group that has exactly one nontrivial proper subgroup. Prove that G is cyclic and |G|=p^2 where p is prime.
    Let H be the non-trivial proper subgroup.
    Then \{ e \} \subset H \subset G.

    Pick  a \in G - H. And construct \left< a \right>.
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    Explain a little further

    Could you please explain your proof a little more. Thanks.
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    Quote Originally Posted by jonnyfive View Post
    Could you please explain your proof a little more. Thanks.
    Note that once you construct \left< a\right> it cannot be H and cannot be \{ e \}. Since that is a subgroup and it has no other proper non-trivial subgroups it follows that \left< a \right> = G. Since the group is cyclic it is isomorphic to \mathbb{Z}_n for some n\geq 1. By the properties of cyclic groups we can show that only n=p^2 are the ones with this unique non-trivial proper subgroup property.
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