What is the significance of a group with no proper subgroups?

[Ackbach]

Here is this week's POTW:

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Suppose $G$ is a group with no proper subgroups. What can be said about $G?$ Prove your statements.

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[Ackbach]

Congratulations to johng for his correct solution to this week's POTW, which follows:

Spoiler

$G$ is finite and cyclic of order 1 or cyclic of order a prime.
Let $E=\langle 1\rangle$ be the identity subgroup of $G$ and suppose $G\neq E$. Let $x\in G$ with $x\neq 1$. Then $\langle x\rangle=G$ and $x$ has finite order, for otherwise $\langle x^2\rangle$ is a proper subgroup of $G$. Suppose $x$ has order $n$ with $n$ composite, say $n=mq$. Then $\langle x^m\rangle$ has order $q$ and so would be a proper subgroup of $G$. Hence $n$ is prime. Thus $G=\langle x\rangle$ is cyclic of prime order $n$.