My Math Forum Sylow subgroup of some factor group.

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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 $[KN/N:PN/N]=\frac{|KN|}{|N|} \frac{|N|}{|PN|}= \frac{|KN|}{|PN|}= \frac{|K||N|}{|K \cap N|} \frac{|P \cap N|}{|P||N|} = \frac{|K||P \cap N|}{|P||K \cap N|}=[K:P] \frac{|P \cap N|}{|K \cap N|}$. 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

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

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