Editing Symmetric group (section)

== Applications ==
The symmetric group on a set of size ''n'' is the [[Galois group]] of the general [[polynomial]] of degree ''n'' and plays an important role in [[Galois theory]].  In [[invariant theory]], the symmetric group acts on the variables of a multi-variate function, and the functions left invariant are the so-called [[symmetric function]]s.  In the [[representation theory of Lie groups]], the [[representation theory of the symmetric group]] plays a fundamental role through the ideas of [[Schur functor]]s.  

In the theory of [[Coxeter group]]s, the symmetric group is the Coxeter group of type A<sub>''n''</sub> and occurs as the [[Weyl group]] of the [[general linear group]].  In [[combinatorics]], the symmetric groups, their elements ([[permutation]]s), and their [[group representation|representations]] provide a rich source of problems involving [[Young tableaux]], [[plactic monoid]]s, and the [[Bruhat order]].  [[Subgroup]]s of symmetric groups are called [[permutation group]]s and are widely studied because of their importance in understanding [[Group action (mathematics)|group action]]s, [[homogeneous space]]s, and [[automorphism group]]s of [[Graph (discrete mathematics)|graph]]s, such as the [[Higman–Sims group]] and the [[Higman–Sims graph]].