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 $SL_{n-1}$ ).
随便看	degrees of freedom DehnsLemma DehnsTheorem DehnSurgery Dehn surgery Dehn’s lemma Dehn’s theorem DelayTheorem delay theorem DeletionOperationOnLanguages deletion operation on languages Delta1Bootstrapping DeltaDistribution delta distribution DeltaSystem delta system DeltaSystemLemma delta system lemma DemloNumber Demlo number DeMoivreIdentity de Moivre identity DeMoivreIdentityProofOf de Moivre identity, proof of DeMorganAlgebra