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.