# 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:

- Order of Symmetric Group: the order of $S_n$ is $n!$

- Results about
**the symmetric groups**can be found here.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.5$. Examples of groups: Example $79$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Example $7.5$ - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.6$: Theorem $5$ - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.13$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 76$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 5$: Groups $\text{I}$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 34$. Examples of groups: $(4)$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Examples of groups $\text{(iii)}$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**symmetric group** - 1992: William A. Adkins and Steven H. Weintraub:
*Algebra: An Approach via Module Theory*... (previous) ... (next): $\S 1.1$ - 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 1.5$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**complete symmetric group** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**permutation group** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**symmetric group**