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If $G$ is a finite group and $x$ is a $p'$ element of G does this imply that there a Hall $p'$ subgroup of $G$ containing $x$?

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No. Take $p=2$, $x=(12345)$, and $G=A_5$. It is true if $G$ is solvable. – user641 Apr 4 '12 at 18:04
Not all groups have any Hall p-prime subgroups. A5 has no 2-prime subgroup (no subgroup of order 15) but it certainly has elements of odd order. – Jack Schmidt Apr 4 '12 at 18:05
Thank you very much. – user28083 Apr 4 '12 at 18:29

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