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** *g***N**=

**N***g*, or (by

multiplying on the left by

*g*^{-1})

**N**=

**N**^{g}=

*g*^{-1}**N***g*.

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.