Editing Symmetric group (section)

== Low degree groups ==
{{See also|Representation theory of the symmetric group#Special cases}}
The low-degree symmetric groups have simpler and exceptional structure, and often must be treated separately.

;S<sub>0</sub> and S<sub>1</sub>: The symmetric groups on the [[empty set]] and the [[singleton set]] are trivial, which corresponds to {{math|1=0! = 1! = 1}}.  In this case the alternating group agrees with the symmetric group, rather than being an index 2 subgroup, and the sign map is trivial. In the case of S<sub>0</sub>, its only member is the [[empty function]].

;S<sub>2</sub>: This group consists of exactly two elements: the identity and the permutation swapping the two points.  It is a [[cyclic group]] and is thus [[abelian group|abelian]].  In [[Galois theory]], this corresponds to the fact that the [[quadratic formula]] gives a direct solution to the general [[quadratic polynomial]] after extracting only a single root.  In [[invariant theory]], the representation theory of the symmetric group on two points is quite simple and is seen as writing a function of two variables as a sum of its symmetric and anti-symmetric parts: Setting {{math|1=''f''<sub>s</sub>(''x'', ''y'') = ''f''(''x'', ''y'') + ''f''(''y'', ''x'')}}, and {{math|1=''f''<sub>a</sub>(''x'', ''y'') = ''f''(''x'', ''y'') − ''f''(''y'', ''x'')}}, one gets that {{math|1=2⋅''f'' = ''f''<sub>s</sub> + ''f''<sub>a</sub>}}.  This process is known as [[symmetrization]].

;S<sub>3</sub>: S<sub>3</sub> is the first nonabelian symmetric group.  This group is isomorphic to the [[dihedral group of order 6]], the group of reflection and rotation symmetries of an [[equilateral triangle]], since these symmetries permute the three vertices of the triangle. Cycles of length two correspond to reflections, and cycles of length three are rotations.  In Galois theory, the sign map from S<sub>3</sub> to S<sub>2</sub> corresponds to the resolving quadratic for a [[cubic polynomial]], as discovered by [[Gerolamo Cardano]], while the A<sub>3</sub> kernel corresponds to the use of the [[discrete Fourier transform]] of order 3 in the solution, in the form of [[Lagrange resolvent]]s.{{citation needed|date=September 2009}}

;S<sub>4</sub>: The group S<sub>4</sub> is isomorphic to the group of proper rotations about opposite faces, opposite diagonals and opposite edges, [[Rencontres numbers|9, 8 and 6]] permutations, of the [[cube]].<ref>{{cite thesis |first=J. |last=Neubüser |title=Die Untergruppenverbände der Gruppen der Ordnungen ̤100 mit Ausnahme der Ordnungen 64 und 96 |date=1967 |type=PhD |publisher=Universität Kiel |url=}}</ref> Beyond the group [[Alternating group|A<sub>4</sub>]], S<sub>4</sub> has a [[Klein four-group]] V as a proper [[normal subgroup]], namely the even transpositions {{math|{(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)},}} with quotient S<sub>3</sub>. In [[Galois theory]], this map corresponds to the resolving cubic to a [[quartic polynomial]], which allows the quartic to be solved by radicals, as established by [[Lodovico Ferrari]]. The Klein group can be understood in terms of the [[Lagrange resolvent]]s of the quartic.  The map from S<sub>4</sub> to S<sub>3</sub> also yields a 2-dimensional irreducible representation, which is an irreducible representation of a symmetric group of degree ''n'' of dimension below {{math|''n'' − 1}}, which only occurs for {{math|1=''n'' = 4}}.

;S<sub>5</sub>: S<sub>5</sub> is the first non-solvable symmetric group.  Along with the [[special linear group]] {{math|SL(2, 5)}} and the [[icosahedral group]] {{math|A<sub>5</sub> × S<sub>2</sub>}}, S<sub>5</sub> is one of the three non-solvable groups of order 120, up to isomorphism.  S<sub>5</sub> is the [[Galois group]] of the general [[quintic equation]], and the fact that S<sub>5</sub> is not a [[solvable group]] translates into the non-existence of a general formula to solve [[quintic polynomial]]s by radicals.  There is an exotic inclusion map {{math|S<sub>5</sub> → S<sub>6</sub>}} as a [[#Transitive subgroup anchor|transitive subgroup]]; the obvious inclusion map {{math|S<sub>''n''</sub> → S<sub>''n''+1</sub>}} fixes a point and thus is not transitive. This yields the outer automorphism of S<sub>6</sub>, discussed below, and corresponds to the resolvent sextic of a quintic.

;S<sub>6</sub>: Unlike all other symmetric groups, S<sub>6</sub>, has an [[outer automorphism]].  Using the language of [[Galois theory]], this can also be understood in terms of [[Lagrange resolvents]]. The resolvent of a quintic is of degree 6—this corresponds to an exotic inclusion map {{math|S<sub>5</sub> → S<sub>6</sub>}} as a transitive subgroup (the obvious inclusion map {{math|S<sub>''n''</sub> → S<sub>''n''+1</sub>}} fixes a point and thus is not transitive) and, while this map does not make the general quintic solvable, it yields the exotic outer automorphism of S<sub>6</sub>—see ''[[Automorphisms of the symmetric and alternating groups]]'' for details.

:Note that while A<sub>6</sub> and A<sub>7</sub> have an exceptional [[Schur multiplier]] (a [[Covering groups of the alternating and symmetric groups|triple cover]]) and that these extend to triple covers of S<sub>6</sub> and S<sub>7</sub>, these do not correspond to exceptional Schur multipliers of the symmetric group.

=== Maps between symmetric groups ===

Other than the trivial map {{math|1=S<sub>''n''</sub> → C<sub>1</sub> ≅ S<sub>0</sub> ≅ S<sub>1</sub>}} and the sign map {{math|S<sub>''n''</sub> → S<sub>2</sub>}}, the most notable homomorphisms between symmetric groups, in order of [[relative dimension]], are:
* {{math|S<sub>4</sub> → S<sub>3</sub>}} corresponding to the exceptional normal subgroup {{math|V < A<sub>4</sub> < S<sub>4</sub>}};
* {{math|S<sub>6</sub> → S<sub>6</sub>}} (or rather, a class of such maps up to inner automorphism) corresponding to the outer automorphism of S<sub>6</sub>.
* {{math|S<sub>5</sub> → S<sub>6</sub>}} as a transitive subgroup, yielding the outer automorphism of S<sub>6</sub> as discussed above.
There are also a host of other homomorphisms {{math|S<sub>''m''</sub> → S<sub>''n''</sub>}} where {{math|''m'' < ''n''}}.