In the area of mathematics known as group theory, the correspondence theorem, sometimes referred to as the fourth isomorphism theorem or the lattice theorem, states that if
N
is a normal subgroup of a group
G
, then there exists a bijection from the set of all subgroups
A
of
G
containing
N
, onto the set of all subgroups of the quotient group
G
/
N
. The structure of the subgroups of
G
/
N
is exactly the same as the structure of the subgroups of
G
containing
N
, with
N
collapsed to the identity element.
Specifically, if
G is a group,
N is a normal subgroup of
G,
G
is the set of all subgroups
A of
G such that
N
⊆
A
⊆
G
, and
N
is the set of all subgroups of
G/N,
then there is a bijective map
ϕ
:
G
→
N
such that
ϕ
(
A
)
=
A
/
N
for all
A
∈
G
.
One further has that if A and B are in
G
, and A' = A/N and B' = B/N, then
A
⊆
B
if and only if
A
′
⊆
B
′
;
if
A
⊆
B
then
|
B
:
A
|
=
|
B
′
:
A
′
|
, where
|
B
:
A
|
is the index of A in B (the number of cosets bA of A in B);
⟨
A
,
B
⟩
/
N
=
⟨
A
′
,
B
′
⟩
,
where
⟨
A
,
B
⟩
is the subgroup of
G
generated by
A
∪
B
;
(
A
∩
B
)
/
N
=
A
′
∩
B
′
, and
A
is a normal subgroup of
G
if and only if
A
′
is a normal subgroup of
G
/
N
.
This list is far from exhaustive. In fact, most properties of subgroups are preserved in their images under the bijection onto subgroups of a quotient group.
More generally, there is a monotone Galois connection
(
f
∗
,
f
∗
)
between the lattice of subgroups of
G
(not necessarily containing
N
) and the lattice of subgroups of
G
/
N
: the lower adjoint of a subgroup
H
of
G
is given by
f
∗
(
H
)
=
H
N
/
N
and the upper adjoint of a subgroup
K
/
N
of
G
/
N
is a given by
f
∗
(
K
/
N
)
=
K
. The associated closure operator on subgroups of
G
is
H
¯
=
H
N
; the associated kernel operator on subgroups of
G
/
N
is the identity.
Similar results hold for rings, modules, vector spaces, and algebras.
