How to show that group of order 760 is not simple?

By Sylow's theorem $n_{19}=20$, and $o(N(P)) = 38$, but how to continue after this?

Thanks for any help


closed as off-topic by Amzoti, Yiyuan Lee, Claude Leibovici, Brian Fitzpatrick, Jack Schmidt Mar 24 '14 at 17:07

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I really cannot understand why people are voting to close this question! In the permutation action of the group by conjugation on its set of $20$ Sylow $19$-subgroups, an element of order $2$ normalizing a $19$-cycle would consist of $9$ transpositions, and hence would be an odd permutation. So intersecting with the alternating group gives a normal subgroup of index $2$.

  • 1
    $\begingroup$ I voted to close as a duplicate. It seems completely on topic to me, and showed more work than the original one. $\endgroup$ – Jack Schmidt Mar 24 '14 at 17:09

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