# Normal subgroup

In

mathematics, a

**normal subgroup** *N* of a

group *G* is a

subgroup invariant under

conjugation; that is, for each element

*x* in

*N* and each

*g* in

*G*, the element

*g*^{-1}xg is still in

*N*. The statement

*N is a normal subgroup of G* is written:

- .

Another way to put this is saying that right and left cosets of

*N* in

*G* coincide:

*N g = g g*^{-1} N g = g N for all *g* in *G*.

Normal subgroups are of relevance because if

*N* is normal, then the

factor group *G*/

*N* may be formed.
Normal subgroups of

*G* are precisely the

kernelss of group homomorphisms

*f* :

*G* `->` *H*.

{*e*} and *G* are always normal subgroups of *G*. If these are the only ones, then *G* is said to be simple.

See also: characteristic subgroup