# Definition:Symmetric Group/Also known as

## Definition

In view of the isomorphism between symmetric groups on sets of the same cardinality, the terminology **symmetric group of degree $n$** is often used when the nature of the underlying set is immaterial.

Some sources use the term **$n$th symmetric group**.

These terms will sometimes be used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Some sources refer to the **symmetric group** on a set as the **full symmetric group (on $S$)**.

Others use the term **complete symmetric group**.

Similarly, the **symmetric group on $n$ letters** can be found referred to as the **full symmetric group on $n$ letters**.

The term **(full) symmetric group on $n$ objects** can be found for both the general **symmetric group** and the **symmetric group on $n$ letters**

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 $A$.

In line with this, the notation $S \paren n$ is often used for $S_n$ to denote the **symmetric group on $n$ letters**.

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

The notation $\operatorname {Sym} \paren n$ for $S_n$ can also be found.

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

Such sources consequently 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.

Be careful not to refer to $\struct {\Gamma \paren S, \circ}$ for $\card S = n$ or $S_n$ as the **symmetric group of order $n$**, as the order of these groups is not $n$ but $n!$, from Order of Symmetric Group.

### Isomorphism between Symmetric Groups

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.

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 34$. Examples of groups: $(4) \ \text {(c)}$