Fourth isomorphism theorem

This article gives the statement, and possibly proof, of a basic fact in group theory.
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Name

This result is termed the lattice isomorphism theorem, the fourth isomorphism theorem, and the correspondence theorem.

Statement

Symbolic statement

Let $G$ be a group and let $N$ be a Normal subgroup (?) of $G$. Then, we have a bijection:

Set of subgroups of $G$ containing $N$ $\leftrightarrow$ Set of subgroups of $G/N$

If $\phi :G\to G/N$ is the quotient map, then this bijection is given by:

$H\mapsto \phi (H)$

in the forward direction, and:

$K\mapsto {\phi }^{-1}(K)$

in the reverse direction. Moreover:

1. Under the bijection, normality is preserved. In other words, a subgroup containing $N$ is normal if and only if its image under $\phi$ is normal.
2. The bijection is an isomorphism between the lattice of subgroups of $G$ containing $N$, and the lattice of subgroups of $G/N$. In other words, the bijection preserves partial order: $A\le B$ if and only if $\phi (A)\le \phi (B)$. It also preserves intersections and joins.
3. The bijection preserves index. If $A,B$ are subgroups of $G$ containing $N$, with $A\le B$, then $[B:A]=[\phi (B):\phi (A)]$.

References

Textbook references

• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 99, Theorem 20, Section 3.3 (few steps of proof given, but full proof not provided)
• Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 75, Exercise 8, Section 7 (Restriction of a homomorphism to a subgroup) (starred problem, termed Correspondence Problem)