Definition:Symmetric Group

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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 $S$).

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.


Also see

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

  • Results about the symmetric groups can be found here.


Sources