Group Theory/Simple groups and Sylow's theorem

Definition (Sylow p-subgroup):

Let $G$ be a group and let $p\in \mathbb {N}$ be a prime number such that $p||G|$. Then a Sylow $p$-subgroup of $G$ is a subgroup $P\leq G$ such that $|P|=p^{k}$, where $k$ is maximal such that $p^{k}||G|$.

Theorem (Cauchy's theorem):

Let $G$ be a group whose order $|G|$ is divisible by a prime number $p\in \mathbb {N}$. Then $G$ contains an element of order $p$.

Proof: $G$ acts on itself via conjugation. Let $x_{1},\ldots ,x_{n}\in G$ be a system of representatives of cojugacy classes. The class equation yields

$|G|=\sum _{k=1}^{n}[G:G_{x_{k}}]$.

Either, there exists $k\in \{1,\ldots ,n\}$ such that $[G:G_{x_{k}}]$ is both not $1$ and not divisible by $p$, in which case we may conclude by induction on the group order, noting that $p$ divides $|G_{x_{k}}|$ and $|G_{x_{k}}|<|G|$, or for all $k\in \{1,\ldots ,n\}$ the number $[G:G_{x_{k}}]$ is either $1$ or divisible by $p$; but in this case, by taking the class equation $\mod p$, we obtain that $Z(G)$ is nontrivial and moreover that its order is divisible by $p$. Hence, it suffices to consider the case where $G$ is an abelian group. Take then any element $h\in G$. If $h$ has order divisible by $p$, raising $h$ to a sufficiently high power will produce an element of order $p$. Otherwise, the order of $G/\langle h\rangle$ is divisible by $p$, and by induction we find an element $g+\langle h\rangle \in G/\langle h\rangle$ whose order is divisible by $p$. Then the order of $g$ will also be divisible by $p$, because otherwise, passing to the quotient, $(g+\langle h\rangle )^{k}=e+\langle h\rangle$ for some $k$ not divisible by $p$. $\Box$

Theorem (Sylow's theorem):

Let $G$ be a finite group, such that $|G|=p^{k}m$ with $p\nmid m$. Then the following hold:

1. $G$ has a Sylow subgroup
2. The action of $G$ by conjugation on the Sylow subgroups is transitive
3. If $n_{p}$ is the number of Sylow $p$-subgroups, then $n_{p}|m$ and $n_{p}\equiv 1\mod p$
4. Every $p$-group of $G$ is contained within some Sylow $p$-group of $G$

Definition (simple group):

A group $G$ is a simple group if $G$ and $\{e\}$ are the only normal subgroups of $G$ (where $e\in G$ denotes the identity).