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Math Help - A group with no subgroups

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    A group with no subgroups

    Suppose that G is a group with more than one element and G has no proper, nontrivial subgroups. Prove that |G| is prime.
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    Super Member Rebesques's Avatar
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    |G| must have no proper divisors, so...?
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    Then the |G| can only be divided by itself or 1. So it is a prime.

    This is easy, too, should have gotten it earlier.

    Thanks.
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    Quote Originally Posted by tttcomrader View Post
    Suppose that G is a group with more than one element and G has no proper, nontrivial subgroups. Prove that |G| is prime.
    Choose x\in G so that x\not = e which is possible by hypothesis. Construct the cyclic subgroup \left< x\right>. By hypothesis this must generate G for this subgroup is nontrivial. Hence, G is isomorphic to the cyclic group \mathbb{Z}_n. Now this group has proper non-trivial subgroups unless n is a prime. Thus, |G|=p.
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