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 $S$.

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 $A$.

However, does not adopt this practice.

Also defined as
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.

Also see

 * Symmetric Group is Group, demonstrating that $\struct {\Gamma \paren S, \circ}$ is indeed a group.

If $S$ is finite with cardinality $n$, then:
 * Number of Permutations: the order of $\struct {\Gamma \paren S, \circ}$ is $n!$
 * Symmetric Group on n Letters is Isomorphic to Symmetric Group: $\struct {\Gamma \paren S, \circ}$ is isomorphic to the symmetric group on $n$ letters