Cayley’s theorem

Let $G$ be a group, then $G$ is isomorphic to a subgroup of the permutation group $S_G$

If $G$ is finite and of order $n$, then $G$ is isomorphic to a subgroup of the permutation group $S_n$

Furthermore, suppose $H$ is a proper subgroup of $G$. Let $X=\{Hg|g\in G\}$ be the set of right cosets in $G$. The map $\theta :G\to S_X$ given by $\theta (x)(Hg)=Hgx$ is a homomorphism. The kernel is the largest normal subgroup of $H$. We note that $|S_X|=[G:H]!$. Consequently if $|G|$ doesn’t divide $[G:H]!$ then $\theta$ is not an isomorphism so $H$ contains a non-trivial normal subgroup, namely the kernel of $\theta$.