In group theory, the correspondence theorem (also the lattice theorem, and variously and ambiguously the third and fourth isomorphism theorem) states that if N {\displaystyle N} is a normal subgroup of a group G {\displaystyle G}, then there exists a bijection from the set of all subgroups A {\displaystyle A} of G {\displaystyle G} containing N {\displaystyle N}, onto the set of all subgroups of the quotient group G / N {\displaystyle G/N}. Loosely speaking, the structure of the subgroups of G / N {\displaystyle G/N} is exactly the same as the structure of the subgroups of G {\displaystyle G} containing N {\displaystyle N}, with N {\displaystyle N} collapsed to the identity element.

Specifically, if

G {\displaystyle G} is a group,

N ◃ G {\displaystyle N\triangleleft G}, a normal subgroup of G {\displaystyle G},

G = { A ∣ N ⊆ A ≤ G } {\displaystyle {\mathcal {G}}=\{A\mid N\subseteq A\leq G\}}, the set of all subgroups A {\displaystyle A} of G {\displaystyle G} that contain N {\displaystyle N}, and

N = { S ∣ S ≤ G / N } {\displaystyle {\mathcal {N}}=\{S\mid S\leq G/N\}}, the set of all subgroups of G / N {\displaystyle G/N},

then there is a bijective map ϕ : G → N {\displaystyle \phi :{\mathcal {G}}\to {\mathcal {N}}} such that

ϕ ( A ) = A / N {\displaystyle \phi (A)=A/N} for all A ∈ G . {\displaystyle A\in {\mathcal {G}}.}

One further has that if A {\displaystyle A} and B {\displaystyle B} are in G {\displaystyle {\mathcal {G}}} then

  • A ⊆ B {\displaystyle A\subseteq B} if and only if A / N ⊆ B / N {\displaystyle A/N\subseteq B/N};
  • if A ⊆ B {\displaystyle A\subseteq B} then | B : A | = | B / N : A / N | {\displaystyle |B:A|=|B/N:A/N|}, where | B : A | {\displaystyle |B:A|} is the index of A {\displaystyle A} in B {\displaystyle B} (the number of cosets b A {\displaystyle bA} of A {\displaystyle A} in B {\displaystyle B});
  • ⟨ A , B ⟩ / N = ⟨ A / N , B / N ⟩ , {\displaystyle \langle A,B\rangle /N=\left\langle A/N,B/N\right\rangle ,} where ⟨ A , B ⟩ {\displaystyle \langle A,B\rangle } is the subgroup of G {\displaystyle G} generated by A ∪ B ; {\displaystyle A\cup B;}
  • ( A ∩ B ) / N = A / N ∩ B / N {\displaystyle (A\cap B)/N=A/N\cap B/N}, and
  • A {\displaystyle A} is a normal subgroup of G {\displaystyle G} if and only if A / N {\displaystyle A/N} is a normal subgroup of G / N {\displaystyle 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 ∗ ) {\displaystyle (f^{*},f_{*})} between the lattice of subgroups of G {\displaystyle G} (not necessarily containing N {\displaystyle N}) and the lattice of subgroups of G / N {\displaystyle G/N}: the lower adjoint of a subgroup H {\displaystyle H} of G {\displaystyle G} is given by f ∗ ( H ) = H N / N {\displaystyle f^{*}(H)=HN/N} and the upper adjoint of a subgroup K / N {\displaystyle K/N} of G / N {\displaystyle G/N} is a given by f ∗ ( K / N ) = K {\displaystyle f_{*}(K/N)=K}. The associated closure operator on subgroups of G {\displaystyle G} is H ¯ = H N {\displaystyle {\bar {H}}=HN}; the associated kernel operator on subgroups of G / N {\displaystyle G/N} is the identity. A proof of the correspondence theorem can be found .

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.

See also