Symmetric group

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols.[1] Since there are ( factorial) such permutation operations, the order (number of elements) of the symmetric group is .

A Cayley graph of the symmetric group S4
Cayley table, with header omitted, of the symmetric group S3. The elements are represented as matrices. To the left of the matrices, are their two-line form. The black arrows indicate disjoint cycles and correspond to cycle notation. Green circle is an odd permutation, white is an even permutation and black is the identity.

These are the positions of the six matrices

Some matrices are not arranged symmetrically to the main diagonal – thus the symmetric group is not abelian.

Although symmetric groups can be defined on infinite sets, 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.

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 is isomorphic to a subgroup of the symmetric group on (the underlying set of) .

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