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

Question

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?

If it had an element of order $120$ it would be cyclic (hence abelian). — lulu, May 09 '16 at 14:12

score 2 · Accepted Answer · answered May 09 '16 at 14:12

2

$S_{5}$ 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?

answered May 09 '16 at 14:12

ervx

1 Answers1