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.