In mathematics, the symmetric group on a set is the group consisting of all automorphisms of the set (all one-to-one and onto functions from the set to itself) with function composition as the group operation.
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 ...
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In mathematics, the symmetric group on a set is the group consisting of all automorphisms of the set (all one-to-one and onto functions from the set to itself) with function composition as the group operation.
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.
The symmetric group on a set X is the group whose underlying set is the collection of all bijections from X to X and whose group operation is that of function composition. The symmetric group of degree n is the...
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