Symmetric group

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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