A
subgroup N of a
group G is called
normal if it has the same left
cosets as right cosets, or equivalently if
forall g in
G gN=
Ng, or (by
multiplying on the left by
g-1)
N=
Ng=
g-1Ng.
In an Abelian group, this last definition obviously holds for any subgroup. However, non-commutative groups may have subgroups which are not normal.
See also example of a normal subgroup of a normal subgroup which is not normal.