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symmetric group (Definition)

Let $ X$ be a set. Let $ S(X)$ be the set of permutations of $ X$ (i.e. the set of bijective functions on $ X$). Then the act of taking the composition of two permutations induces a group structure on $ S(X)$. We call this group the symmetric group and it is often denoted $ {\rm Sym}(X)$.

When $ X$ has a finite number $ n$ of elements, we often refer to the symmetric group as $ S_n$, and describe the elements by using cycle notation.



"symmetric group" is owned by antizeus.
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See Also: symmetry


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symmetric group on three letters (Example) by Wkbj79
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Cross-references: cycle notation, finite, structure, group, composition, bijective functions, permutations
There are 25 references to this entry.

This is version 2 of symmetric group, born on 2003-11-10, modified 2003-11-29.
Object id is 5421, canonical name is SymmetricGroup2.
Accessed 7942 times total.

Classification:
AMS MSC: 20B30 (Group theory and generalizations :: Permutation groups :: Symmetric groups)
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Duplicate by bbukh on 2003-11-11 00:09:24
This entry is exact word-by-word duplicate of http://planetmath.org/encyclopedia/SymmetricGroup.html.

Boris
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