Symmetric Groups of Same Order are Isomorphic/Proof 1

Theorem

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $T_1$ and $T_2$ be sets whose cardinality $\card {T_1}$ and $\card {T_2}$ are both $n$.

Let $\struct {\map \Gamma {T_1}, \circ}$ and $\struct {\map \Gamma {T_2}, \circ}$ be the symmetric group on $S$ and $T$ respectively.

Then $\struct {\map \Gamma {T_1}, \circ}$ and $\struct {\map \Gamma {T_2}, \circ}$ are isomorphic.

Proof

Consider the symmetric group on $n$ letters $S_n$.

From Symmetric Group on n Letters is Isomorphic to Symmetric Group we have that:

$\struct {\map \Gamma {T_1}, \circ}$ is isomorphic to $S_n$

$\struct {\map \Gamma {T_2}, \circ}$ is isomorphic to $S_n$

and hence from Isomorphism is Equivalence Relation:

$\struct {\map \Gamma {T_1}, \circ}$ is isomorphic to $\struct {\map \Gamma {T_2}, \circ}$.

$\blacksquare$