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.

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