Groupprops, The Group Properties Wiki (pre-alpha)

YOUR FEEDBACK IS IMPORTANT!

Please take a short user satisfaction survey about Groupprops.

Your survey responses will be helpful in improving the site experience!

Thanks in advance!

From Groupprops

Contents 1 Name 2 Statement 2.1 Statement with symbols 3 References 3.1 Textbook references

This article gives the statement, and possibly proof, of a basic fact in group theory.
View a complete list of basic facts in group theory
VIEW FACTS USING THIS: directly | directly or indirectly, upto two steps | directly or indirectly, upto three steps|

This article is about an isomorphism theorem in group theory.
View a complete list of isomorphism theorems| Read a survey article about the isomorphism theorems

Name

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

Statement

Statement with symbols

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 Set of subgroups of G / N

If is the quotient map, then this bijection is given by:

in the forward direction, and:

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 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: if and only if . It also preserves intersections and joins.
3. The bijection preserves index. If A,B are subgroups of G containing N, with , then .

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)