Show $S_3$, the permutation group on three letters, is not cyclic.

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Show $S_3$, the permutation group on three letters, is not cyclic.

I don't know where to start. Cyclic groups confuse me, permutation groups more so.

Any help/hints would be most welcome.

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2 Answers

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This group is: $$S_3=\{ id, (2,3), (1,2), (1,2,3), (1,3,2), (1,3) \}$$ of order $6$. Now consider tow elements of it, for example, $x=(1,2)$ and $y=(2,3)$. We have: $$yx=(1,2,3)\neq (1,3,2)=xy$$ So the group is not abelian and so it cannot be cyclic.

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$S_3$ is non-abelian. ${}{}{}{}$

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