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In class, one way we're taught to prove a group is not simple is to exhibit a normal Sylow subgroup.

I'm wondering if the converse is true, i.e. if a group is not simple, must it have a normal Sylow subgroup? I can't seem to prove this is the case but I haven't been able to come up with a counterexample yet either.

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For a counterexample, consider $S_n$, $n\geq 5$. No $p$-subgroup is normal as the only proper nontrivial normal subgroup is the alternating group.

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    $\begingroup$ In fact no Sylow subgroup of $S_4$ is normal. I think that's the smallest group with that property. $\endgroup$ – Derek Holt May 14 '15 at 16:49
  • $\begingroup$ @DerekHolt I wasn't sure about that and was too lazy to try to check so I took the easy way out. Thanks for the info. $\endgroup$ – Matt Samuel May 14 '15 at 16:51
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Counterexample: if $S$ is a non-abelian simple group, then certainly $S \times S$ is not simple and has no normal Sylow subgroups.

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