D+F 4.5.34

Let $P\in{Syl_p(G)}$ and let $N \unlhd G$.

(a) Use the conjugacy part of Sylow's Theorem to prove that $P \cap N\in{Syl_p(N)}$

(b) Prove that $PN/N\in{Syl_p(G/N)}$

The second part of Sylow's Theorem states: Let $P$ be a Sylow p-subgroup of $G$ and $Q$ a p-subgroup of $G$. Then, $\exists g\in{G}$ s.t. $Q \leq gPg^{-1}$, i.e. $Q$ is contained in some conjugate of $P$.

$\textit{Proof}$ : Let $|G|=p^am$ where $p \nmid m$. $|N|$ divides $|G|$ by Lagrange, so let $|N|=p^bn$ where $b \leq a$, $p \nmid n$ and $n \leq m$. To show that $P \cap N \in{Syl_p(N)}$ we need to show that $|P \cap N|=p^b$.
Observe that $P \cap N$ is a p-group: $$|P \cap N| \mid |P|=p^a$$ $$|P \cap N| \mid |N|=p^bn$$ Since $p$ is prime, $$\Longrightarrow |P \cap N|=p^c \quad (c \leq b)$$ We want to now show that $b \leq c$.
Corollary 15, p. 94 states that if $H,K \leq G$ and $H \leq N_G(K)$ then the subset product, $HK \leq G$. In this case, $N \lhd G \Longrightarrow N_G(N) = G$, so the subset product $PN \leq G$. By the 4th isomorphism theorem, we may mod out by $N$, so we get that $$PN/N \leq G/N \quad (\star)$$ The order of the subset product is given by the formula $$|PN| = \frac{|P| \cdot |N|}{|P \cap N|} \Longrightarrow \frac{|PN|}{|N|}=\frac{|P|}{|P \cap N|}$$ Combining with $(\star)$ we get $$\frac{|P|}{|P \cap N|} \mid \frac{|G|}{|N|}=\frac{p^am}{p^bn}=p^{a-b}\frac{m}{n}$$ $$\frac{p^a}{p^c} \mid p^{a-b}\frac{m}{n}$$ $p^{a-c}$ is a power of a prime and $p^{a-b}$ is a power of a prime, so $$p^{a-c} \mid p^{a-b} \Longrightarrow a-c \leq a-b \Longrightarrow b \leq c$$ Thus, $ |P \cap N| = p^b \quad \therefore P \cap N \in{Syl_p(N)}$.

To prove that $PN/N\in{Syl_p(G/N)}$, we make a similar argument based on subgroup orders. $$|G/N|=\frac{|G|}{|N|}=\frac{p^am}{p^bn}=p^{a-b}(\frac{m}{n})$$ Where we can say that $\frac{m}{n}$ is in lowest terms and is a positive integer.
Again, the order of subset products is given by $$|PN|=\frac{|P| \cdot |N|}{|P \cap N|}$$ Rearranging, we get $$|PN/N|=\frac{|P|}{|P \cap N|}=\frac{p^a}{p^b}=p^{a-b}$$ $$\Longrightarrow PN/N \in{Syl_p(G/N)} \quad \blacksquare$$