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March 30th, 2016, 12:13 PM   #1
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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
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