# 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 the term **symmetric group of degree $n$**.

Some sources use $\map S n$ 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.

## Motivation

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 $S_n = \set {1, 2, \ldots, n}$ of cardinality $n$ is selected, and the symmetric group is defined on $S_n$ 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.

## Also see

- 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): $\S 7$: Example $7.5$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Examples of Group Structure: $\S 30$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 5$: Groups $\text{I}$ - 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 34$. Examples of groups: $(4) \ \text {(c)}$ - 1992: William A. Adkins and Steven H. Weintraub:
*Algebra: An Approach via Module Theory*... (previous) ... (next): $\S 1.1$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $9$: Permutations