Does an injective homomorphism always exists from $G$ into $GL_n(R)$ where order of $G$ is $n$?

Question

We have a group $G$ of finite order $n$. Does a one to one homomorphism always exist from $G$ to general linear group?

Answer 1

Yes, because by Cayley's theorem, $G$ can be considered as a subgroup of $S_n$ (by $G \rightarrow \rm{Sym}(G), g \mapsto (h \mapsto gh)$.

But $S_n$ acts on the standard basis $(e_1, \dots, e_n)$ of $\Bbb{R}^n$ in the natural way (i.e. $\sigma e_i = e_{\sigma^{-1}(i)}$).

By extending each map $\sigma$ acting on the set $\{e_1, \dots, e_n \}$ to a unique(!), invertible(!) linear map, we get the desired homomorphism.