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Prove that if $P$ is a $p$-Sylow of $G$ and $N \triangleleft\> G$ then:

  1. $ PN/N $ is q $p$-Sylow of $G/N$

  2. $P \cap N $ is $p$-Sylow of $N$

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up vote 2 down vote accepted

Better than writing this out, you better read this excellent expository paper by Keith Conrad, Corollary 7.5 and Theorem 7.7.

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