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 functions. In the representation theory of Lie groups, the representation theory of the symmetric group plays a fundamental role through the ideas of Schur functors. In the theory of Coxeter groups, the symmetric group is the Coxeter group of type An and occurs as the Weyl group of the general linear group. In combinatorics, the symmetric groups, their elements (permutations), and their representations provide a rich source of problems involving Young tableaux, plactic monoids, and the Bruhat order. Subgroups of symmetric groups are called permutation groups and are widely studied because of their importance in understanding group actions, homogeneous spaces, and automorphism groups of graphs, such as the Higman–Sims group and the Higman–Sims graph. Group properties and special elements The elements of the symmetric group on a set X are the permutations of X. Multiplication The group operation in a symmetric group is function composition, denoted by the symbol ∘ or simply by just a composition of the permutations. The composition of permutations f and g, pronounced "f of g", maps any element x of X to f(g(x)). Concretely, let (see permutation for an explanation of notation): Applying f after g maps 1 first to 2 and then 2 to itself; 2 to 5 and then to 4; 3 to 4 and then to 5, and so on. So composing f and g gives A cycle of length , taken to the kth power, will decompose into k cycles of length m: For example, (, ), Verification of group axioms To check that the symmetric group on a set X is indeed a group, it is necessary to verify the group axioms of closure, associativity, identity, and inverses. The operation of function composition is closed in the set of permutations of the given set X. Function composition is always associative. The trivial bijection that assigns each element of X to itself serves as an identity for the group. Every bijection has an inverse function that undoes its action, and thus each element of a symmetric group does have an inverse which is a permutation too. Transpositions, sign, and the alternating