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Let $G$ be a group. Suppose that $G$ has at most two nontrivial subgroups. Show that $G$ is cyclic.


Can anyone help me please to solve the problem?

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If $G$ has no nontrivial subgroups, it is clearly cyclic. If $G$ has exactly one nontrivial subgroup $H$, consider the subgroup generated by a nonidentity element $g\in G\setminus H$.

Now suppose that $H$ and $K$ are the only nontrivial subgroups of $G$. Recall that a group is never the union of two proper subgroups. So pick some element $g\in G\setminus (H\cup K)$. What must $\langle g\rangle$ be?

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  • $\begingroup$ If G has no nontrivial subgroups, it is clearly cyclic.But why? $\endgroup$
    – gumti
    Jun 25, 2013 at 15:43
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    $\begingroup$ @gumti If $G$ is trivial, you're done. Otherwise, pick a nonidentity element of $g$ of $G$. What must the subgroup $\langle g\rangle$ be? Remember, it cannot be nontrivial! $\endgroup$
    – Ben West
    Jun 25, 2013 at 15:45
  • $\begingroup$ "Recall that a group is never the union of two proper subgroups." $\endgroup$
    – Myself
    Jun 25, 2013 at 16:28

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