Editing Cayley's theorem (section)

=== Alternative setting of proof ===
An alternative setting uses the language of [[Group action (mathematics)|group action]]s. We consider the group <math>G</math> as acting on itself by left multiplication, i.e. <math>g \cdot x = gx</math>, which has a permutation representation, say <math>\phi : G \to \mathrm{Sym}(G)</math>.

The representation is faithful if <math>\phi</math> is injective, that is, if the kernel of <math>\phi</math> is trivial. Suppose <math>g\in\ker\phi</math>. Then, <math>g = ge = g\cdot e = e</math>. Thus, <math>\ker\phi</math> is trivial. The result follows by use of the [[first isomorphism theorem]], from which we get <math>\mathrm{Im}\, \phi \cong G</math>.