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March 30th, 2016, 12:13 PM  #1 
Newbie Joined: Aug 2012 Posts: 5 Thanks: 0  Sylow subgroup of some factor group.
Hi. Let $G$ be a finite group. Let $K$ be a subgroup of $G$ and let $N$ be a normal subgroup of $G$. Let $P$ be a Sylow $p$subgroup of $K$. Is $PN/N$ is a Sylow $p$subgroup of $KN/N$? Here is what I think. Since $PN/N \cong P/(P \cap N)$, then $PN/N$ is a $p$subgroup of $KN/N$. Now $[KN/N:PN/N]=\frac{KN}{N} \frac{N}{PN}= \frac{KN}{PN}= \frac{KN}{K \cap N} \frac{P \cap N}{PN} = \frac{KP \cap N}{PK \cap N}=[K:P] \frac{P \cap N}{K \cap N}$. Since $P$ is a Sylow $p$subgroup of $K$, then $p$ does not divide $[K:P]$. Also, $p$ does not divide $\frac{P \cap N}{K \cap N}$ as $\frac{P \cap N}{K \cap N} \leq 1$ because $P \cap N$ is a subgroup of $K \cap N$. Therefore $p$ does not divide $[KN/N:PN/N]$. Thus $PN/N$ is a Sylow $p$subgroup of $KN/N$. Am I right? Thanks in advance 

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abstract algebra, factor, group, group theory, subgroup, sylow 
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