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Let $A$ be an abelian and transitive subgroup of $S_n$ ($A$ acts on $\left\{ 1,2,\ldots,n\right\}$ naturally). Prove that $A$ is cyclic and generated by cycle permutation of length $n$.

I've proved that $\left|A\right|=n$, but couldn't show $A$ contains such cycle..


marked as duplicate by Dietrich Burde, Namaste group-theory Dec 12 '14 at 12:52

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  • $\begingroup$ Saw this, didn't find my answer there $\endgroup$ – daPollak Dec 12 '14 at 12:21
  • $\begingroup$ What makes you think this is true? $\endgroup$ – Derek Holt Dec 12 '14 at 12:30
  • $\begingroup$ @DietrichBurde Why if $A$ has no normal subgroups than it is cyclic ? $\endgroup$ – daPollak Dec 12 '14 at 12:42
  • $\begingroup$ I will check what this comment then means - sorry. Here is the link. $\endgroup$ – Dietrich Burde Dec 12 '14 at 12:52
  • $\begingroup$ It says that the subgroup in question can contain no normal subgroup (of the whole group). But the subgroup in question here is the trivial subgroup, so the statement does not say very much at all! $\endgroup$ – Derek Holt Dec 12 '14 at 12:54

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