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 {gN : g in G} (this is a partition of G). The operation on G/N is given by (gN)*(hN) = (ghN). 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.