The nth Symmetric group is the group of permutations of the set X={1,2,...,n} with binary operation given by composition of functions. It is denoted Sn.

A permutation is simply a bijection X->X. Thus Sn has n ! elements.

We can write the elements of Sn as products of disjoint cycles. A cycle is written like this
(a1 a2 ... at)
and this denotes the permutation that maps a1 to a2, a2 to a3, etc etc and at to a1. The other elements of X are fixed by this cycle. Sn has a normal subgroup called the Alternating group, and denoted An consisting of all the even permutations. These are the permutations that can be written as a product of an even number of transpositions.

S3 is the first non-abelian group. It has elements (in cycle notation)
{ 1,(12),(13),(23),(123),(132)}
It has a normal subgroup A3={1,(123),(132)}.

See also permutation group.

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