Let

**G** be a

group, and let

**N** be a

normal subgroup of

**G**. We define the

*quotient group* **G**/

**N** ("

**G** up to **N**", or "

**G** modulo **N**") to have the

set of elements {

*g***N** :

*g* in

**G**} (this is a

partition of

**G**). The

operation on

**G**/

**N** is given by (

*g***N**)*(

*h***N**) = (

*gh***N**). Using the fact that

**N** is normal, we may prove that this operation is

well defined (i.e. does not depend on the choice of representatives

*g*,

*h* for the 2 elements of

**G**/

**N**). It then follows easily that this is a group.

Note that a quotient group is generally *not* a subgroup!

The isomorphism theorems give various connections between quotient groups, homomorphisms, and normal subgroups.