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Prove that $S_5$ has no elements of order $120$.

I know that by Lagrange's theorem, elements can have order $1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120$.

I feel like I need to use some sort of contradiction argument here but I'm not sure where to go with that. Any suggestions?


marked as duplicate by Dietrich Burde, user296602, user147263, user228113, Chris Godsil May 10 '16 at 0:42

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    $\begingroup$ If it had an element of order $120$ it would be cyclic (hence abelian). $\endgroup$ – lulu May 9 '16 at 14:12

$S_{5}$ has order $120$. If there was an element of order $120$ in $S_{5}$, it would mean $S_{5}$ is a cyclic group. Is $S_{5}$ cyclic?


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