Group Theory

For any finite group $G$, the order (number of elements) of every subgroup $H$ of $G$ divides the order of $G$.

Let $\phi: G \to H$ be a group homomorphism. Then the kernel of $\phi$ is a normal subgroup of $G$, and the image of $\phi$ is isomorphic to the quotient group $G / \ker(\phi)$.

If $G$ is a finite group and $p^n$ divides the order of $G$ (where $p$ is prime), then $G$ has a subgroup of order $p^n$.

Every group $G$ is isomorphic to a subgroup of the symmetric group acting on $G$.