Quotient of Sylow P-Subgroup
Theorem
Let $P$ be a Sylow $p$-subgroup of a finite group $G$.
Let $N$ be a normal subgroup of $G$.
Then $P N / N$ is a Sylow $p$-subgroup of $G / N$.
Proof
As $P \leq G$ and $ N \triangleleft G$, we have $P N / N \cong P / \left({P \cap N}\right)$ by the Second Isomorphism Theorem.
Note that $P N$ is a $p$-subgroup of $G / N$.
From Intersection of Normal Subgroup with Sylow P-Subgroup,we have that $p \nmid \left[{G : P N}\right]$.
So $P N / N$ is a Sylow $p$-subgroup of $G / N$.
Note that $P N$ is a $p$-subgroup of $G / N$.
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Sources
- John F. Humphreys: A Course in Group Theory (1996): $\S 11$: Proposition $11.14 \ \text{(ii)}$
