Editing Symmetric group (section)

== Group properties and special elements ==
The elements of the symmetric group on a set ''X'' are the [[permutation]]s of ''X''.

=== Multiplication ===
The group operation in a symmetric group is function composition, denoted by the symbol ∘ or by simple juxtaposition. The composition {{math|''f'' ∘ ''g''}} of permutations {{mvar|f}} and {{mvar|g}}, pronounced "{{mvar|f}} of {{mvar|g}}", maps any element {{mvar|x}} of {{mvar|X}} to {{math|''f''(''g''(''x''))}}. Concretely, let (see [[permutation]] for an explanation of notation):
<math display=block> f = (1~3)(2)(4~5)=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 1 & 5 & 4\end{pmatrix},</math>
<math display=block> g = (1~2~5)(3~4)=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1\end{pmatrix}.</math>
Applying {{mvar|f}} after {{mvar|g}} maps 1 first to 2 and then 2 to itself; 2 to 5 and then to 4; 3 to 4 and then to 5, and so on. So, composing {{mvar|f}} and {{mvar|g}} gives
<math display=block> fg = f\circ g = (1\ 2\ 4)(3\ 5)=\begin{pmatrix} 1 & 2 &3 & 4 & 5 \\ 2 & 4 & 5 & 1 & 3\end{pmatrix}.</math>

A [[Cyclic permutation|cycle]] of length {{math|''L'' {{=}} ''k'' · ''m''}}, taken to the {{mvar|k}}th power, will decompose into {{mvar|k}} cycles of length {{mvar|m}}: For example, ({{math|''k'' {{=}} 2}}, {{math|''m'' {{=}} 3}}),
<math display=block> (1~2~3~4~5~6)^2 = (1~3~5) (2~4~6).</math>

=== Verification of group axioms ===
To check that the symmetric group on a set ''X'' is indeed a [[group (mathematics)|group]], it is necessary to verify the group axioms of closure, associativity, identity, and inverses.<ref>{{cite book |last1=Vasishtha |first1=A.R. |last2=Vasishtha |first2=A.K. |chapter=2. Groups S3 Group Definition |chapter-url={{GBurl|45eCTUS6YnQC|p=49}} |title=Modern Algebra |publisher=Krishna Prakashan Media |date=2008 |isbn=9788182830561 |pages=49 }}</ref>
# The operation of function composition is closed in the set of permutations of the given set ''X''.
# Function composition is always associative.
# The trivial bijection that assigns each element of ''X'' to itself serves as an identity for the group.
# Every bijection has an [[inverse function]] that undoes its action, and thus each element of a symmetric group does have an inverse which is a permutation too.

=== Transpositions, sign, and the alternating group ===
{{main article|Transposition (mathematics)}}
A '''transposition''' is a permutation which exchanges two elements and keeps all others fixed; for example (1 3) is a transposition. Every permutation can be written as a product of transpositions; for instance, the permutation ''g'' from above can be written as ''g'' = (1 2)(2 5)(3 4). Since ''g'' can be written as a product of an odd number of transpositions, it is then called an [[Even and odd permutations|odd permutation]], whereas ''f'' is an even permutation.

The representation of a permutation as a product of transpositions is not unique; however, the number of transpositions needed to represent a given permutation is either always even or always odd. There are several short proofs of the invariance of this parity of a permutation.

The product of two even permutations is even, the product of two odd permutations is even, and the product of one of each is odd.  Thus we can define the '''sign''' of a permutation:

:<math>\operatorname{sgn}f = \begin{cases} +1, & \text{if }f\mbox { is even} \\ -1, & \text{if }f \text{ is odd}. \end{cases}</math>

With this definition,
:<math>\operatorname{sgn}\colon \mathrm{S}_n \rightarrow \{+1, -1\}\ </math>
is a [[group homomorphism]] ({+1, −1} is a group under multiplication, where +1 is e, the [[neutral element]]). The [[Kernel (algebra)|kernel]] of this homomorphism, that is, the set of all even permutations, is called the '''[[alternating group]]''' A<sub>''n''</sub>. It is a [[normal subgroup]] of S<sub>''n''</sub>, and for {{nowrap|''n'' ≥ 2}} it has {{nowrap|''n''!/2}} elements. The group S<sub>''n''</sub> is the [[semidirect product]] of A<sub>''n''</sub> and any subgroup generated by a single transposition.

Furthermore, every permutation can be written as a product of ''[[adjacent transposition]]s'', that is, transpositions of the form {{nowrap|(''a'' ''a''+1)}}.  For instance, the permutation ''g'' from above can also be written as {{nowrap|1=''g'' = (4 5)(3 4)(4 5)(1 2)(2 3)(3 4)(4 5)}}. The sorting algorithm [[bubble sort]] is an application of this fact.  The representation of a permutation as a product of adjacent transpositions is also not unique.

=== Cycles ===
A [[cyclic permutation|cycle]] of ''length'' ''k'' is a permutation ''f'' for which there exists an element ''x'' in {1, ..., ''n''} such that ''x'', ''f''(''x''), ''f''<sup>2</sup>(''x''), ..., ''f''<sup>''k''</sup>(''x'') = ''x'' are the only elements moved by ''f''; it conventionally is required that {{nowrap|''k'' ≥ 2}} since with {{nowrap|1=''k'' = 1}} the element ''x'' itself would not be moved either. The permutation ''h'' defined by
: <math>h = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 2 & 1 & 3 & 5\end{pmatrix}</math>
is a cycle of length three, since {{nowrap|1=''h''(1) = 4}}, {{nowrap|1=''h''(4) = 3}} and {{nowrap|1=''h''(3) = 1}}, leaving 2 and 5 untouched. We denote such a cycle by {{nowrap|(1 4 3)}}, but it could equally well be written {{nowrap|(4 3 1)}} or {{nowrap|(3 1 4)}} by starting at a different point. The order of a cycle is equal to its length. Cycles of length two are transpositions. Two cycles are ''disjoint'' if they have disjoint subsets of elements. Disjoint cycles [[Commutative property|commute]]: for example, in S<sub>6</sub> there is the equality {{nowrap|1=(4 1 3)(2 5 6) = (2 5 6)(4 1 3)}}. Every element of S<sub>''n''</sub> can be written as a product of disjoint cycles; this representation is unique [[up to]] the order of the factors, and the freedom present in representing each individual cycle by choosing its starting point.

Cycles admit the following conjugation property with any permutation <math>\sigma</math>, this property is often used to obtain its [[Symmetric group#Generators and relations|generators and relations]].

:<math>\sigma\begin{pmatrix} a & b & c & \ldots \end{pmatrix}\sigma^{-1}=\begin{pmatrix}\sigma(a) & \sigma(b) & \sigma(c) & \ldots\end{pmatrix}</math>

=== Special elements ===
Certain elements of the symmetric group of {1, 2, ..., ''n''} are of particular interest (these can be generalized to the symmetric group of any finite totally ordered set, but not to that of an unordered set).

The '''{{visible anchor|order reversing permutation}}''' is the one given by:
:<math>\begin{pmatrix} 1 & 2 & \cdots & n\\
n & n-1 & \cdots & 1\end{pmatrix}.</math>
This is the unique maximal element with respect to the [[Bruhat order]] and the
[[longest element of a Coxeter group|longest element]] in the symmetric group with respect to generating set consisting of the adjacent transpositions {{nowrap|(''i'' ''i''+1)}}, {{nowrap|1 ≤ ''i'' ≤ ''n'' − 1}}.

This is an involution, and consists of <math>\lfloor n/2 \rfloor</math> (non-adjacent) transpositions
:<math>(1\,n)(2\,n-1)\cdots,\text{ or }\sum_{k=1}^{n-1} k = \frac{n(n-1)}{2}\text{ adjacent transpositions: }</math>
:: <math>(n\,n-1)(n-1\,n-2)\cdots(2\,1)(n-1\,n-2)(n-2\,n-3)\cdots,</math>

so it thus has sign:

:<math>\mathrm{sgn}(\rho_n) = (-1)^{\lfloor n/2 \rfloor} =(-1)^{n(n-1)/2} = \begin{cases}
+1 & n \equiv 0,1 \pmod{4}\\
-1 & n \equiv 2,3 \pmod{4}
\end{cases}</math>
which is 4-periodic in ''n''.

In S<sub>2''n''</sub>, the ''[[Faro shuffle|perfect shuffle]]'' is the permutation that splits the set into 2 piles and interleaves them. Its sign is also <math>(-1)^{\lfloor n/2 \rfloor}.</math>

Note that the reverse on ''n'' elements and perfect shuffle on 2''n'' elements have the same sign; these are important to the classification of [[Clifford algebra]]s, which are 8-periodic.