# Let $G$ be a finite group, $H$ be a $s$-permutable subgroup of $G$. if index $|G:H|=p$. then $H$ is normal subgroup of $G$.

$H$ 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..

• Hint: First show that $H$ contains all $q$-Sylow subgroups for $q\neq p$. – Tobias Kildetoft Jun 18 '16 at 8:35
• @TobiasKildetoft I show that $H$ contains all $q$-Sylow subgroups for $q \neq p$.can you explain more. thanks – sina Jun 18 '16 at 17:27
• The subgroup generated by those Sylow subgroups is normal with index a power of $p$. – Tobias Kildetoft Jun 18 '16 at 17:42
• @TobiasKildetoft can you give me the reference? – sina Jun 18 '16 at 18:18
• 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. – Tobias Kildetoft Jun 18 '16 at 18:40