symmetric group

Let $X$ be a set.Let ${\\rm Sym}(X)$ be the set of permutations of $X$ (i.e. the set of bijective functions from $X$ to itself).Then the act of taking the composition of two permutationsinduces a group structure on ${\\rm Sym}(X)$ .We call this group the symmetric group.

The group ${\\rm Sym}(\\{1,2,\\ldots,n\\})$ is often denoted $S_{n}$ or $\\mathfrak{S}_{n}$ .

$S_{n}$ is generated by the transpositions $\\{(1,2),(2,3),\\ldots,(n-1,n)\\}$ ,and by any pair of a 2-cycle and $n$ -cycle.

$S_{n}$ is the Weyl group of the $A_{n-1}$ root system (and hence of the special linear group $SL_{n-1}$ ).

单词	SymmetricGroup
释义	symmetric group Let $X$ be a set.Let ${\\rm Sym}(X)$ be the set of permutations of $X$ (i.e. the set of bijective functions from $X$ to itself).Then the act of taking the composition of two permutationsinduces a group structure on ${\\rm Sym}(X)$ .We call this group the symmetric group. The group ${\\rm Sym}(\\{1,2,\\ldots,n\\})$ is often denoted $S_{n}$ or $\\mathfrak{S}_{n}$ . $S_{n}$ is generated by the transpositions $\\{(1,2),(2,3),\\ldots,(n-1,n)\\}$ ,and by any pair of a 2-cycle and $n$ -cycle. $S_{n}$ is the Weyl group of the $A_{n-1}$ root system (and hence of the special linear group $SL_{n-1}$ ).
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