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: $$S_n$$.

Some sources refer to this as the full symmetric group (on $$n$$ letters).

$$\left({S_n, \circ}\right)$$ 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 $$I_{S_n}$$ 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 $$\left|{T}\right| = n$$, $$\left({S_n, \circ}\right) \cong \left({\Gamma \left({T}\right), \circ}\right)$$.

Also see
Some sources refer to the Group of Permutations of any general set $$S$$ with $$n$$ elements as the (full) symmetric group on $$S$$.

It matters little, as (by this result) all such groups are isomorphic anyway.

However, it can sometimes be convenient to be able to refer unambiguously to the contents of this group by using cycle notation (or indeed, two-row notation if you really want to) without confusion. If all such definitions are based on an underlying set containing a rigorously specified set of natural numbers, this makes certain aspects of this discipline significantly easier.

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