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.

By definition of cardinality, as $$\left|{T}\right| = n$$ we can find a bijection between $$T$$ and $$\N_n$$.

From Number of Permutations, it is immediate that $$\left|{\left({\Gamma \left({T}\right), \circ}\right)}\right| = n! = \left|{\left({S_n, \circ}\right)}\right|$$.

Again, we can find a bijection $$\phi$$ between $$\left({\Gamma \left({T}\right), \circ}\right)$$ and $$\left({S_n, \circ}\right)$$.

The result follows directly from the Transplanting Theorem.

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

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

It can of course be convenient sometimes 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.

We can stretch the definition for countable $$S$$, as in that case there is a bijection between $$S$$ and $$\N$$ by definition of countability.

However, this definition can not apply if $$S$$ is uncountable.

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

Others use $$\mathcal S_n$$ or some such variant.

Some older sources denote the symmetric group on $$n$$ letters as $$\mathfrak S_n$$.

However, the fraktur font is rarely used nowadays as it is cumbersome to reproduce and awkward to read.