Definition:Symmetric Group

Theorem
Let $$S_n$$ denote the set of permutations on $n$ letters.

The structure $$\left({S_n, \circ}\right)$$, where $$\circ$$ denotes composition of mappings, forms a group.

This is called the Symmetric Group on $$n$$ Letters, and is usually denoted, when the context is clear, without the operator: $$S_n$$.

$$\left({S_n, \circ}\right)$$ is isomorphic to the Group of Permutations of the $$n\,$$ elements of any set $$T$$ whose cardinality is $$n$$.

That is, $$\forall T \subseteq \mathbb{U}, \left|{T}\right| = n: \left({S_n, \circ}\right) \cong \left({\Gamma \left({T}\right), \circ}\right)$$.

In order not to make notation overly cumbersome, the product notation is usually used for composition, thus $$\pi \circ \rho$$ is written $$\pi \rho$$.

Also, for the same reason, rather than using $$I_{\mathbb{N}_n}$$ for the identity mapping, the symbol $$e$$ is usually used.

Proof
The fact that $$\left({S_n, \circ}\right)$$ is a group follows directly from Group of Permutations.

Now we need to show that, for any given $$T$$ such that $$\left|{T}\right| = n$$, $$\left({S_n, \circ}\right) \cong \left({\Gamma \left({T}\right), \circ}\right)$$.

Notation
Some sources use $$S \left({n}\right)$$ for $$S_n$$.