Quotient of Sylow P-Subgroup
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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
We have that $P \le G$ and $ N \lhd G$.
So by the Second Isomorphism Theorem for Groups:
- $P N / N \cong P / \paren {P \cap N}$
We have that:
- $P N / N = \set {p N : p \in P}$
and so every element of $P N / N$ has order a power of $p$.
Hence $P N / N$ is a $p$-subgroup of $G / N$.
From Intersection of Normal Subgroup with Sylow P-Subgroup, we have that:
- $p \nmid \index G {P N}$
So $P N / N$ is a Sylow $p$-subgroup of $G / N$.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $11$: The Sylow Theorems: Proposition $11.14 \ \text{(ii)}$