Definition:Symmetric Group

Definition
Let $S$ be a set.

Let $\Gamma \paren S$ be the set of permutations on $S$.

Let $\struct {\Gamma \paren S, \circ}$ be the algebraic structure such that $\circ$ denotes the composition of mappings.

Then $\struct {\Gamma \paren S, \circ}$ is called the symmetric group on $S$.

If $S$ has $n$ elements, then $\struct {\Gamma \paren S, \circ}$ is often denoted $S_n$.

Notation
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.

Also known as
Some sources refer to this as the full symmetric group (on $S$).

Some sources use the notation $S \paren A$ to denote the set of permutations on a given set $A$, and thence $S \paren A$ to denote the symmetric group on $A$.

Some older sources denote the symmetric group on $A$ as $\mathfrak S_A$.

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

Also defined as
Some sources, in recognition that Symmetric Groups of Same Order are Isomorphic, take the view that it is unimportant to distinguish rigorously between symmetric groups on different sets.

Hence they refer to a symmetric group on a set of cardinality $n$ as the $n$th symmetric group, and denote it $S_n$ or a variant.

It is convenient to refer to the elements of $S_n$ using cycle notation (or indeed, two-row notation if you really want to).

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.

Also see

 * Symmetric Group is Group


 * Symmetric Groups of Same Order are Isomorphic

If $S$ is finite with cardinality $n$, then:
 * Order of Symmetric Group: the order of $S_n$ is $n!$