Symmetric Group on n Letters is Isomorphic to Symmetric Group

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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:

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

where:

$\map \Gamma T$ denotes the set of permutations on $T$


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$\struct {\map \Gamma T, \circ}$ denotes an algebraic structure with $1$ operation
$\circ$ denotes composition of mappings.


Proof

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.

$\blacksquare$


Sources

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