Editing Symmetric group (section)

== Definition and first properties ==
The symmetric group on a finite set <math>X</math> is the group whose elements are all bijective functions from <math>X</math> to <math>X</math> and whose group operation is that of [[function composition]].<ref name=Jacobson-def /> For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group of '''degree''' <math>n</math> is the symmetric group on the set <math>X = \{1, 2, \ldots, n\}</math>.

The symmetric group on a set <math>X</math> is denoted in various ways, including <math>\mathrm{S}_X</math>, <math>\mathfrak{S}_X</math>, <math>\Sigma_X</math>, <math>X!</math>, and <math>\operatorname{Sym}(X)</math>.<ref name=Jacobson-def />  If <math>X</math> is the set <math>\{1, 2, \ldots, n\}</math> then the name may be abbreviated to <math>\mathrm{S}_n</math>, <math>\mathfrak{S}_n</math>, <math>\Sigma_n</math>, or <math>\operatorname{Sym}(n)</math>.<ref name=Jacobson-def />

Symmetric groups on infinite sets behave quite differently from symmetric groups on finite sets, and are discussed in {{harv|Scott|1987|loc=Ch. 11}}, {{harv|Dixon|Mortimer|1996|loc=Ch. 8}}, and {{harv|Cameron|1999}}.

The symmetric group on a set of <math>n</math> elements has [[order (group theory)|order]] <math>n!</math> (the [[factorial]] of <math>n</math>).<ref>{{harvnb|Jacobson|2009|p=32 Theorem 1.1}}</ref> It is [[abelian group|abelian]] if and only if <math>n</math> is less than or equal to 2.<ref>{{cite web|title=Symmetric Group is not Abelian/Proof 1|url=https://proofwiki.org/wiki/Symmetric_Group_is_not_Abelian/Proof_1}}</ref> For <math>n=0</math> and <math>n=1</math> (the [[empty set]] and the [[singleton set]]), the symmetric groups are [[trivial group|trivial]] (they have order <math>0! = 1! = 1</math>). The group S<sub>''n''</sub> is [[solvable group|solvable]] if and only if <math>n \leq 4</math>. This is an essential part of the proof of the [[Abel–Ruffini theorem]] that shows that for every <math>n > 4</math> there are [[polynomial]]s of degree <math>n</math> which are not solvable by radicals, that is, the solutions cannot be expressed by performing a finite number of operations of addition, subtraction, multiplication, division and root extraction on the polynomial's coefficients.