The

*n*th

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

*S*_{n}.

A permutation is simply a bijection *X->X*.
Thus *S*_{n} has *n !* elements.

We can write the elements of *S*_{n}
as products of disjoint cycles. A cycle is written like this

*(a*_{1} a_{2} ... a_{t})

and this denotes the permutation that maps
*a*_{1} to *a*_{2},
*a*_{2} to *a*_{3},
etc etc and *a*_{t} to *a*_{1}.
The other elements of *X* are fixed by this cycle.
*S*_{n} has a normal subgroup
called the Alternating group, and denoted *A*_{n} consisting of all the even permutations.
These are the permutations that can be written as a product of
an even number of transpositions.

*S*_{3} is the first non-abelian group. It has
elements (in cycle notation)

{ 1,(12),(13),(23),(123),(132)}

It has a normal subgroup *A*_{3}={1,(123),(132)}.

See also permutation group.