Difference between revisions of "Symmetric group"
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| − | The '''symmetric group''' <math>S_{n}</math> is defined to be the [[group]] of all [[permutation]]s of <math>n</math> objects. | + | The '''symmetric group''' <math>S_{n}</math> is defined to be the [[group]] of all [[permutation]]s of <math>n</math> objects. In this context, a permutation is to be thought of as a [[bijective]] [[function]] from a [[set]] of size <math>n</math> to itself, and the group operation is [[composition]] of functions. |
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| + | The symmetric group is of interest in many different branches of mathematics, especially [[combinatorics]]. (This is in part due to the various different representations of permutations: as functions, as products of [[cycle]]s, as [[sequence]]s or [[word]]s, etc.) Knowledge of the symmetric group <math>S_{n}</math> can also be crucial in such areas as [[Galois theory]]. For example, an important theorem in [[Galois theory]] is that the [[Galois group]] of the general polynomial equation of degree <math>n</math> is <math>S_{n}</math>, and this can be used to prove that [[polynomial]] [[equation]]s of degree five and higher are unsolvable through the use of elementary arithmetic and root extractions. | ||
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Revision as of 14:53, 15 March 2008
The symmetric group 
is defined to be the group of all permutations of 
objects. In this context, a permutation is to be thought of as a bijective function from a set of size to itself, and the group operation is composition of functions.
The symmetric group is of interest in many different branches of mathematics, especially combinatorics. (This is in part due to the various different representations of permutations: as functions, as products of cycles, as sequences or words, etc.) Knowledge of the symmetric group can also be crucial in such areas as Galois theory. For example, an important theorem in Galois theory is that the Galois group of the general polynomial equation of degree is , and this can be used to prove that polynomial equations of degree five and higher are unsolvable through the use of elementary arithmetic and root extractions.
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