Symmetric group

related topics
{math, number, function}
{group, member, jewish}
{style, bgcolor, rowspan}

In mathematics, the symmetric group on a set is the group consisting of all bijections of the set (all one-to-one and onto functions) from the set to itself with function composition as the group operation.[1]

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 isomorphic to a subgroup of the symmetric group on G.

This article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a 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.

Contents

Full article ▸

related documents
Golomb coding
Euler's totient function
Search algorithm
Division algebra
Constant of integration
Exact sequence
Kruskal's algorithm
Mersenne prime
Convex set
Separable space
Probability space
Analytic geometry
Local ring
Mersenne twister
Group representation
Controllability
Carmichael number
Goodstein's theorem
Banach fixed point theorem
Inverse limit
Linear equation
Automated theorem proving
Well-order
Diophantine set
Lebesgue measure
Kolmogorov space
Laurent series
Cardinality
Cartesian product
Topological group