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Let $G$ be a group $\lvert\,G\,\rvert = 14$ acting on a set $M$ of $7$ elements without fixed points. Prove that the action is transitive and that $G$ is isomorphic to $D_7$ or $\mathbb{Z}/14\mathbb{Z}$. I proved that the action is transitive but I can't prove the second part, I don't know even how to start! Any help is appreciated, thanks!

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The second part has been proved at this site using various methods. In fact, any group of order $2p$, for $p>2$ prime is isomorphic to either $C_{2p}$ or $D_p$. For $p=7$ we obtain your result.

References:

Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .

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