# Definition:Symmetric Group

## Definition

Let $S$ be a set.

Let $\map \Gamma S$ denote the set of permutations on $S$.

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

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

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

### Symmetric Group on $n$ Letters

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.

## Notation

In order not to make notation for operations on a symmetric group overly cumbersome, product notation is usually used for mapping 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

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 $\SS_n$ 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.

## Also see

If $S$ is finite with cardinality $n$, then:

• Results about the symmetric groups can be found here.