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Can a non-cyclic group of order 8 act transitively on a set of 8 elements? From the orbit-stabilizer theorem I see that the transitive group action of a group of order 8 on a set of 8 elements is necessarily free but I do not see how to conclude that the group has to be cyclic.

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  • $\begingroup$ Furthermore, a cyclic group does not need to act transitively. Take, for example the cyclic group generated by $(1,2)(3,4,5)$ (which has order 6). $\endgroup$
    – ahulpke
    Commented Sep 4, 2019 at 23:41

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Yes, if the order of $G$ is $8$, the action of $G$ on $G$ defined by $L_g(x)=gx$ is transitive. Every group acts transitively on itself.

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