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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 Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post limes5 Abstract Algebra 2 December 31st, 2013 06:06 AM envision Abstract Algebra 1 October 4th, 2009 03:24 AM Spartan Math Abstract Algebra 6 September 21st, 2009 09:52 PM Erdos32212 Abstract Algebra 0 December 8th, 2008 06:15 PM weier Abstract Algebra 1 November 20th, 2006 10:40 AM

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