Definition:Symmetric Group/n Letters

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

Let $\struct {S_n, \circ}$ denote the symmetric group on $S_n$.

Then $\struct {S_n, \circ}$ is referred to as the symmetric group on $n$ letters.

It is often denoted, when the context is clear, without the operator: $S_n$.

Also known as
Some sources refer to this as the full symmetric group on $n$ letters.

Some sources use $S \paren n$ or $\operatorname {Sym} \paren n$ 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, this fraktur font is rarely used nowadays as it is cumbersome to reproduce and awkward to read.

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 on $n$ Letters is Isomorphic to Symmetric Group