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Homesymmetric group

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# 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 permutations induces 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 $SL_{{n-1}}$).

Related:

Group, Cycle2, CayleyGraphOfS_3, Symmetry2

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

20B30*no label found*

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## Comments

## Hmm

I swear that I would have corrected this object had I known about the correction. There are no notices in my mailbox about the correction, or warnings about becoming orphaned. Strange.