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This question already has an answer here:

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?

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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
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$S_{5}$ 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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