Through three minds: The Permutation Representation Theorem

Verse I: The act, the play, the vision~

Cayley's Act:
Cayley's theorem- a profound result in the theory of groups was proven in his 1854 paper titled On The Theory of Groups, which states that 'every finite group is a subgroup of a symmetric group', however he considered only faithful actions on the group.

Jordan's Play:

Camille Jordan in his work Traité des Substitutions, formalized permutation groups and extended Cayley's ideas by closely observing non-faithful actions and thereby determining that homomorphisms need not be injective and hence Cayley's idea was then extended to actions on arbitrary sets.

Noether's Vision:
Emmy Noether- a name not unknown, worked futher on this idea and remoulded it in terms of abstract thinking. The theorem was then reframed as a universal property of group actions, much influenced by her axiomatic reasoning. 

Verse II: The statement and the proof~
Theorema
The actions of $G$ on a set $X$ are in fact, the group homomorphisms from $G$ to $Sym(X)$.

Probationem
Consider a group $G$ acting on a non-empty set $X$, and the group action $\phi$ is defined using the left action as, $\phi : G \times X \rightarrow X$  such that  $\phi (g,x)= g \cdot x$

Now, what this map does is ­­­­take an element from the set $X$ and map it to some element in this set itself .
This is simply a permutation on the set $X$ and hence, this map would then lie in $Sym(X).$
That is, $\phi \in Sym(X)$
Now, consider elements $g_1, g_2\in G$ then, $$\phi(g_1, \phi (g_2, x))= \phi(g_1, g_2 \cdot x)= g_1 \cdot (g_2 \cdot (x))= (g_1 \cdot g_2) \cdot (x)= \phi(g_1 g_2) \cdot (x)$$ and, $\phi (e, x)= e \cdot x= x$ (the identity map).
Now,  $\phi (g^{-1})$ exists, and the composition of $\phi(g)$ and $\phi(g^{-1})$ will be $\phi(e) .$
Hence, $g \mapsto \phi(g)$ is the homomorphism $G \rightarrow Sym(X)$.

Conversely, consider the homomorphism $f_g :G \rightarrow Sym(X)$ defined as $f_g (x)= g \cdot x$.

Here, $f_g$ is the permutation associated to $g$ under the homomorphism.
The stated map satisfies the axioms for the group actions, i.e., the existence of an identity and fulfillment of the associative property.

For the identity element $e \in G, \  e \cdot x = f_e (x) =e \cdot x= x$
(since $f$ is a homomorphism and must map the identity of $G$ to the identity permutation)

Now, for $g_1, g_2 \in G$ and $x \in X,$ 

$(g_1 g_2) \cdot x = f_{g_1 g_2}(x) = (f_{g_1} \circ f_{g_2}(x) )= f_{g_1}(f_{g_2}(x))
= g_1 \cdot (g_2 \cdot x)$

Here we have used the homomorphism property $\psi(g_1 g_2) = \psi(g_1) \circ \psi(g_2)$.

Thus, every homomorphism $f_g: G \to \text{Sym}(X)$ induces a valid group action.

According to this map, for each $g \in G$, we have a permutation $f_g \in Sym(X).$

We therefore conclude that the actions of a group on a set are essentially the group homomorphisms from the group to the group of permutations of the set.

Much obliged to K. Conrad's papers, which always come to the rescue.