- #1

moont14263

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Let [itex]G[/itex] be a finite group. Let [itex]K[/itex] be a subgroup of [itex]G[/itex] and let [itex]N[/itex] be a normal subgroup of [itex]G[/itex]. Let [itex]P[/itex] be a Sylow [itex]p[/itex]-subgroup of [itex]K[/itex]. Is [itex]PN/N[/itex] is a Sylow [itex]p[/itex]-subgroup of [itex]KN/N[/itex]?

Here is what I think.

Since [itex]PN/N \cong P/(P \cap N)[/itex], then [itex]PN/N[/itex] is a [itex]p[/itex]-subgroup of [itex]KN/N[/itex].

Now [itex][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|}=[/itex] [itex][K:P]\frac{|P \cap N|}{|K \cap N|}[/itex]

Since [itex]P[/itex] is a Sylow [itex]p[/itex]-subgroup of [itex]K[/itex], then [itex]p[/itex] does not divide [itex][K:P][/itex]. Also, [itex]p[/itex] does not divide [itex]\frac{|P \cap N|}{|K \cap N|}[/itex] as [itex]\frac{|P \cap N|}{|K \cap N|} \leq 1[/itex] because [itex]P \cap N[/itex] is a subgroup of [itex]K \cap N[/itex]. Therefore [itex]p[/itex] does not divide [itex][KN/N:PN/N][/itex].

Thus [itex]PN/N[/itex] is a Sylow [itex]p[/itex]-subgroup of [itex]KN/N[/itex].

Am I right?

Thanks in advance