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Suppose a group $G$ has no proper subgroups (that is, the only subgroup of $G$ is $G$ itself and the trivial subgroup $\{e\}$. Show that $G$ is cyclic.

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  • $\begingroup$ Take any $g\in G$ different from the identity element. What is $\langle g\rangle$? $\endgroup$ – user121880 Aug 31 '14 at 23:17
  • $\begingroup$ Let $x\in G$, then $\langle x\rangle=G$ unless $x=e$. $\endgroup$ – Adam Hughes Aug 31 '14 at 23:17
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Hint: For any $g\in G$, we have $\langle g\rangle\subseteq G$, where $\langle g \rangle$ is the cyclic group generated by $g$.

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