Let $H$ and $P$ be subgroups of a finite group $G$. Show that the sets $H x P, x \in G$, 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

$$\frac{|H x P|}{|P|}=\frac{|H|}{\left|H \cap x P x^{-1}\right|}$$

for any $x \in G$. Deduce that if $G$ has a Sylow $p$-subgroup, then so does $H$.

Let $p, n \in \mathbb{N}$ with $p$ a prime. Write down the order of the group $G L_{n}(\mathbb{Z} / p \mathbb{Z})$. Identify in $G L_{n}(\mathbb{Z} / p \mathbb{Z})$ a Sylow $p$-subgroup and a subgroup isomorphic to the symmetric group $S_{n}$. Deduce that every finite group has a Sylow $p$-subgroup.

State Sylow's theorem on the number of Sylow $p$-subgroups of a finite group.

Let $G$ be a group of order $p q$, where $p>q$ are prime numbers. Show that if $G$ is non-abelian, then $q \mid p-1$.