Intuition behind Cayley theorem

Question

Cayley theorem says every group is isomorphic to a group of permutations.

How do I understand this?

I have thought it through group actions. When Permuatation group acts on a set of objects it just permutes those objects. Similarly if any group acts on a set of objects it just permuates them.

Is my intuition correct?

Answer 1

What you have said is not really more than (half of) the definition of a group action.

Cayley's Theorem states that a group $G$ acts faithfully on a very specific set in a natural way, namely on its underlying set. The action is defined as the group multiplication. Here, an action $G \to \mathrm{Sym}(X)$ is called faithful when it is injective.

Answer 2

Originally, people realized that the set $S_X$ of all the bijections on a given set $X$, endowed with map composition as "combining rule", fulfils the properties of closure, associativity, existence of the identity map, existence of the inverse map (one for each bijection). Later on, someone thought: what if we replace the set $S_X$ by any set $G$, endowed with a "combining rule" among its elements (so that the closure property is natively embedded in this abstractization process), such that associativity and indentity/inverses existence are ensured by axioms? We get so the definition of "abstract group", which is then the result of a typical mimicking process. Of course, being the prototypical one, $S_X$ is also an abstract group. The next step was realizing that two abstract groups are algebraically the same when an isomorphism exists between them. Well, Cayley theorem proves that $S_G$ contains an isomorphic copy of any abstract group you can build up with the elements of the set $G$.

Therefore, in a sense, Cayley theorem closes the hystorical loop: "group of permutations" $\rightarrow$ "abstract group" $\rightarrow$ "group of permutations".