2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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 Sep 4 '19 at 23:41
4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.