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*is still in

^{-1}xg*N*. The statement

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

- .

*N*in

*G*coincide:

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

*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