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

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

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

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Better than writing this out, you better read this excellent expository paper by Keith Conrad, Corollary 6.5 and Theorem 6.7.

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