A subgroup N of a group G is a normal subgroup if the left coset
gN and the right coset Ng are equal as sets for each 
.
For a given 
, this does not mean that ng =gn. Rather, it means
that for a given , there is an 
such that ng = gm.
Lots of examples of normal subgroups exist since every subgroup of an abelian group is a normal subgroup. For a non-abelian example, consider the subgroup of rotations in the symmetry group of the regular n-gon. This is a normal subgroup and although we could prove it by calculation it follows from the following result.
Any subgroup N of index 2 in a group G is a normal subgroup of G.
In the group 
the subgroup consisting of the identity element and the
rotation through 
is a normal subgroup.