Symmetric Group on n Letters is Isomorphic to Symmetric Group

Theorem

The symmetric group on $n$ letters $\struct {S_n, \circ}$ is isomorphic to the symmetric group on the $n$ elements of any set $T$ whose cardinality is $n$.

That is:

$\forall T \subseteq \mathbb U, \card T = n: \struct {S_n, \circ} \cong \struct {\Gamma \paren T, \circ}$

The fact that $\struct {S_n, \circ}$ is a group is a direct implementation of the result Symmetric Group is Group.

By definition of cardinality, as $\card T = n$ we can find a bijection between $T$ and $\N_n$.

From Number of Permutations, it is immediate that $\order {\paren {\Gamma \paren T, \circ} } = n! = \order {\struct {S_n, \circ} }$.

Again, we can find a bijection $\phi$ between $\struct {\Gamma \paren T, \circ}$ and $\struct {S_n, \circ}$.

The result follows directly from the Transplanting Theorem.

$\blacksquare$