In group theory, the correspondence theorem[1][2][3][4][5][6][7][8] (also the lattice theorem,[9] and variously and ambiguously the third and fourth isomorphism theorem[6][10]) states that if
is a normal subgroup of a group
, then there exists a bijection from the set of all subgroups
, onto the set of all subgroups of the quotient group
Loosely speaking, the structure of the subgroups of
collapsed to the identity element.
Specifically, if then there is a bijective map
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
between the lattice of subgroups of
: the lower adjoint of a subgroup
and the upper adjoint of a subgroup
The associated closure operator on subgroups of
; the associated kernel operator on subgroups of
A proof of the correspondence theorem can be found here.
Similar results hold for rings, modules, vector spaces, and algebras.
More generally an analogous result that concerns congruence relations instead of normal subgroups holds for any algebraic structure.