Definition:Symmetric Group/Isomorphism

Definition

In recognition that Symmetric Groups of Same Order are Isomorphic, it is unimportant to distinguish rigorously between symmetric groups on different sets.

Hence a representative set of cardinality $n$ is selected, usually (as defined here) $\N^*_{\le n} = \set {1, 2, \ldots, n}$.

The symmetric group $S_n$ is then defined on $\N^*_{\le n}$, and identified as the $n$th symmetric group.

As a consequence, results can be proved about the symmetric group on $n$ letters which then apply to all symmetric groups on sets with $n$ elements.

It is then convenient to refer to the elements of $S_n$ using cycle notation or two-row notation as appropriate.

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 cannot apply if $S$ is uncountable.