$H$ is called $s$-permutable in $G$ if it permutes with every Sylow subgroup of $G$.

Let $G$ be a finite group, $H$ be a $s$-permutable subgroup of $G$. if index $|G:H|=p$, where $p$ is an odd prime divisor of $|G|$, (Not necessarily the smallest prime). then $H$ is normal subgroup of $G$.

Any help/answer/hint will be appreciated so much; thank you all..

  • 1
    $\begingroup$ Hint: First show that $H$ contains all $q$-Sylow subgroups for $q\neq p$. $\endgroup$ – Tobias Kildetoft Jun 18 '16 at 8:35
  • $\begingroup$ @TobiasKildetoft I show that $H$ contains all $q$-Sylow subgroups for $q \neq p$.can you explain more. thanks $\endgroup$ – sina Jun 18 '16 at 17:27
  • $\begingroup$ The subgroup generated by those Sylow subgroups is normal with index a power of $p$. $\endgroup$ – Tobias Kildetoft Jun 18 '16 at 17:42
  • $\begingroup$ @TobiasKildetoft can you give me the reference? $\endgroup$ – sina Jun 18 '16 at 18:18
  • $\begingroup$ This will be in pretty much any book on group theory that goes just a bit beyond the basics. It is also not that hard to show. $\endgroup$ – Tobias Kildetoft Jun 18 '16 at 18:40

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