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. | ||
| − | Knowledge of the general symmetric group <math>S_{n}</math> is crucial in such areas as [[Galois Theory]], including proving that [[polynomial]] [[equation]]s of degree five and higher are unsolvable through the use of elementary arithmetic and root extractions. | + | Knowledge of the general symmetric group <math>S_{n}</math> is crucial in such areas as [[Galois Theory]], including proving that [[polynomial]] [[equation]]s of degree five and higher are unsolvable through the use of elementary arithmetic and root extractions. 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>. |
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Revision as of 15:10, 14 March 2008
The symmetric group 
is defined to be the group of all permutations of 
objects.
Knowledge of the general symmetric group is crucial in such areas as Galois Theory, including proving that polynomial equations of degree five and higher are unsolvable through the use of elementary arithmetic and root extractions. An important theorem in Galois Theory is that the Galois group of the general polynomial equation of degree is .
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