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.
2026-03-27 10:47:56.1774608476
A transitive action of a non-cyclic group of order 8 on a set of 8 elements
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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.