# 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.