PlanetMath: symmetric group

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symmetric group

(Definition)

Let be a set. Let ${\rm Sym}(X)$ be the set of permutations of (i.e. the set of bijective functions from 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 or $\mathfrak{S}_n$ .

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

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

"symmetric group" is owned by bwebste. [ full author list (2) | owner history (1) ]

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See Also: group, cycle, Cayley graph of $S_3$

, symmetry

Attachments:: two isomorphic groups (Example) by Wkbj79
symmetric group is generated by adjacent transpositions (Theorem) by rspuzio

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Cross-references: special linear group, root system, Weyl group, transpositions, generated by, structure, group, composition, bijective functions, permutations
There are 28 references to this entry.

This is version 7 of symmetric group, born on 2001-11-25, modified 2004-12-12.
Object id is 1040, canonical name is SymmetricGroup.
Accessed 6652 times total.

Classification:

AMS MSC:

20B30 (Group theory and generalizations :: Permutation groups :: Symmetric groups)

Pending Errata and Addenda

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