Subgroups and cyclic groups [duplicate]

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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.

marked as duplicate by Adriano, Adam Hughes, user147263, Moishe Kohan, PVAL-inactiveSep 1 '14 at 0:27

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