Consider the following question:
Let $H$ and $P$ be subgroups of a finite group $G$. Show that the sets $HxP$, $x \in G$, partition $G$. By considering the action of $H$ on the set of left cosets of $P$ in $G$ by left multiplication, or otherwise, show that $$\frac{|HxP|}{|P|} = \frac{|H|}{|H \cap xPx^{-1}|} $$ for any $x \in G$. Deduce that if $G$ has a Sylow $p$-subgroup, then so does $H$.
I was able to get to the required equality by considering the Orbit-Stabiliser Theorem and a suitable homomorphism. I am struggling to deduce the last part. What has me confused is that I do not even think its true. For example, $C_{15}$ has a subgroup of order 5. Clearly it also has a Sylow $3$ group. However, the subgroup of size $5$ can not have a Sylow $3$ group. Am I misunderstanding?
I would appreciate it if someone can clarify this for me and also explain the last deduction.