In mathematics, the symmetric group Sn on a finite set of n symbols is the group (mathematics)|group whose elements are all the permutations of the n symbols, and whose group operation is the function composition|composition of such permutations, which are treated as bijection|bijective functions from the set of symbols to itself. Since there are factorial|n! (n factorial) possible permutations of a set of n symbols, it follows that the Order (group theory)|order (the number of elements) of the symmetric group Sn is n!.
Although symmetric groups can be defined on infinite sets as well, this article discusses only the finite symmetric groups: their applications, their elements, their conjugacy classes, a finitely presented group|finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, symmetric group will mean a symmetric group on a finite set.
The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics. Cayley's theorem states that every group G is group isomorphism|isomorphic to a subgroup of the symmetric group on G.
In mathematics, the symmetric group Sn on a finite set of n symbols is the group (mathematics)|group whose elements are all the permutations of the n symbols, and whose group operation is the function composition|composition of such permutations, which are treated as bijection|bijective functions from the set of symbols to itself. Since there are factorial|n! (n factorial) possible permutations of a set of n symbols, it follows that the Order (group theory)|order (the number of elements) of the symmetric group Sn is n!.
Although symmetric groups can be defined on infinite sets as well, this article discusses only the finite symmetric groups: their applications, their elements, their conjugacy classes, a finitely presented group|finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, symmetric group will mean a symmetric group on a finite set.
The symmetric group is important to diverse areas of mathematics...