A finite group is isomorphic to a subgroup of $GL_n(\mathbb Z)$ I'm trying to prove that any finite group $G$ of order $n\ge 2$ is isomorphic to a subgroup of $GL_n(\mathbb Z)$.
My thoughts:
By Cayley, $G$ is isomorphic to a subgroup of $S_n$. Now I need to establish somehow a connection between $S_n$ and $GL_n(\mathbb Z)$. In the case of $n$ small, say $n=2$, $S_3$ acts on $\{e_1,e_2,e_1+e_2\}$, and this action is faithful, which gives an injective homomorphism $GL_2(\mathbb Z)\to S_3$. But a) I guess I need a homomorphism to the other direction (in this case this is an isomorphism, but for larger $n$ it is not I believe) and b) I don't know how to generalize this for larger $n$. 

Any finite group embeds into symmetric group (Cayley's theorem), and symmetric group embeds into $GL_n(Z)$ through permutation matrices.
