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If $G$ has no proper subgroup, prove that $G$ is cyclic of order $p$, where $p$ is a prime number.

I know that since $G $is a group with no proper subgroups, $g \in G$ is not just the identity. I don't know where to go from there.

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    What's the subgroup generated by $g$? What does it mean for that subgroup to not be proper? How does its order relate to the presence of other subgroups?2011-12-15
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    This is already true (and not any harder) if we don't assume that $G$ is finite.2011-12-15
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    for $g\in G$, $\langle g\rangle$ is a cyclic subgroup. the subgroups of a cyclic subgroup of order $n$ are cyclic of order $d|n$. youre condition is then $G=\langle g\rangle$ and order of $g$ equal to $p$ for some prime $p$ (or the trivial group)2011-12-15

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Examine the cyclic subgroup generated by some $g \in G$, where $g$ is not the identity.

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    @user12691: $\langle g \rangle$ is the group generated by $g$, so it must contain the identity element $1$ to be a group. It contains all the powers of $g$. So $\langle g \rangle = \{ 1, g, g^2, \cdots, g^{n-1} \}$, because we're talking about finite groups here (and so the order of $g$ is finite, in this case $n$).2011-12-15
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You don't need to assume $G$ is finite.

Proposition. If $G$ is a group which has no nontrivial proper subgroups, then either $G$ is the trivial group or $G$ is cyclic of prime order.

Proof. If $G$ is the trivial group, we are done. If $G$ is not the trivial group, let $g\in G$ be any element other than the identity. Then $\langle g\rangle$ is a nontrivial subgroup of $G$, and therefore must equal all of $G$ by hypothesis. Thus, $G$ is cyclic.

If $g^2=1$, then $\langle g\rangle =\{1,g\} = G$, so $G$ is cyclic of order $2$ and we are done. If $g^2\neq 1$, then $\langle g^2\rangle$ is a nontrivial subgroup of $G$, so $G=\langle g^2\rangle = \langle g\rangle$, hence there exists $k$ such that $g = (g^2)^k$. Thus, $g^{2k-1}=1$, which proves that $g$ is of finite order. Thus, $G$ is finite cyclic.

Let $n$ be the order of $g$. If $a|n$, $0\lt a\lt n$, then $\langle g^a\rangle = G$ (since it is a nontrivial subgroup). Therefore, $g\in \langle g^a\rangle$, so there exists b such that $g = (g^{a})^b = g^{ab}$. Therefore, $g^{ab-1} = 1$, so $n|ab-1$. Since $a|n$, then $a|ab-1$, hence $a|-1$, so $a=\pm 1$.

That is, the only divisors of $n$ are $\pm 1$ and $\pm n$, so $n$ is prime. $\Box$

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    I don't understand why you said that g^2 ≠g and then ⟨g^2⟩=⟨g⟩2011-12-16
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    That should be $g^2\neq 1$; if $g^2\neq 1$, then $\langle g^2\rangle$ is a subgroup of $G$ that is not trivial, so it must equal $G$; but $G$ already equals $\langle g\rangle$, so $\langle g^2\rangle = \langle g \rangle$.2011-12-16
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    @user12691 Note that I am not asserting that $g^2\neq 1$ necessarily is true. Rather, *either* $g^2=1$ **or** $g^2\neq 1$ (by the law of the excluded middle). If $g^2=1$, then $G$ is cyclic of order $2$ and we are done, and if $g^2\neq 1$, then we continue the argument.2011-12-16
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In view of the original poster's request for further explanation after an answer that got five up-votes, here are further comments.

Let $g$ differ from the identity $e$. Look at $g, g^2, g^3, \ldots$. Since the group is finite, eventually you reach $g^m=\text{some earlier term in this sequence}= g^\ell$. So $\ell

That gets you a cyclic group; now you need to prove that $n$ is prime. Suppose it's not, so that $n=jk$ and $j, k$ are smaller numbers than $n$. Then $g^k, g^{2k}, g^{3k},\ldots,g^{jk}=e$ is a subgroup. But there are no proper subegroups, so the assumption that $n$ is not prime is refuted.

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    The 5 up-voted response wasn't actually an answer. Thank you for your help.2011-12-16
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    @user12691: It was a pointed hint at how *you* could answer your own question.2011-12-16
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HINT $\ $ Any noncyclic group has a proper subgroup generated by any non-identity element. Any infinite cyclic group $\rm\:\left\:$ has the proper subgroup $\rm\:\left\:.\:$ A finite cyclic group $\rm\:\left\:$ of composite order $\rm\:nk\:,\ n,k > 1\:,\:$ has proper subgroup $\rm\:\left\:.\:$ What remains are cyclic groups of prime order.