# Correspondence theorem (group theory)

In the area of mathematics known as group theory, the **correspondence theorem**,[1][2][3][4][5][6][7][8] sometimes referred to as the **fourth isomorphism theorem**[6][9][note 1][note 2] or the **lattice theorem**,[10] states that if is a normal subgroup of a group , then there exists a bijection from the set of all subgroups of containing , onto the set of all subgroups of the quotient group . The structure of the subgroups of is exactly the same as the structure of the subgroups of containing , with collapsed to the identity element.

Specifically, if

*G*is a group,*N*is a normal subgroup of*G*,- is the set of all subgroups
*A*of*G*such that , and - is the set of all subgroups of
*G/N*,

then there is a bijective map such that

- for all

One further has that if *A* and *B* are in , and *A' = A/N* and *B' = B/N*, then

- if and only if ;
- if then , where is the index of
*A*in*B*(the number of cosets*bA*of*A*in*B*); - where is the subgroup of generated by
- , and
- is a normal subgroup of if and only if is a normal subgroup of .

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 between the lattice of subgroups of (not necessarily containing ) and the lattice of subgroups of : the lower adjoint of a subgroup of is given by and the upper adjoint of a subgroup of is a given by . The associated closure operator on subgroups of is ; the associated kernel operator on subgroups of is the identity.

Similar results hold for rings, modules, vector spaces, and algebras.

## See also

## Notes

- Some authors use "fourth isomorphism theorem" to designate the Zassenhaus lemma; see for example by Alperin & Bell (p. 13) or Robert Wilson (2009).
*The Finite Simple Groups*. Springer. p. 7. ISBN 978-1-84800-988-2. - Depending how one counts the isomorphism theorems, the correspondence theorem can also be called the 3rd isomorphism theorem; see for instance H.E. Rose (2009), p. 78.

## References

- Derek John Scott Robinson (2003).
*An Introduction to Abstract Algebra*. Walter de Gruyter. p. 64. ISBN 978-3-11-017544-8. - J. F. Humphreys (1996).
*A Course in Group Theory*. Oxford University Press. p. 65. ISBN 978-0-19-853459-4. - H.E. Rose (2009).
*A Course on Finite Groups*. Springer. p. 78. ISBN 978-1-84882-889-6. - J.L. Alperin; Rowen B. Bell (1995).
*Groups and Representations*. Springer. p. 11. ISBN 978-1-4612-0799-3. - I. Martin Isaacs (1994).
*Algebra: A Graduate Course*. American Mathematical Soc. p. 35. ISBN 978-0-8218-4799-2. - Joseph Rotman (1995).
*An Introduction to the Theory of Groups*(4th ed.). Springer. pp. 37–38. ISBN 978-1-4612-4176-8. - W. Keith Nicholson (2012).
*Introduction to Abstract Algebra*(4th ed.). John Wiley & Sons. p. 352. ISBN 978-1-118-31173-8. - Steven Roman (2011).
*Fundamentals of Group Theory: An Advanced Approach*. Springer Science & Business Media. pp. 113–115. ISBN 978-0-8176-8301-6. - Jonathan K. Hodge; Steven Schlicker; Ted Sundstrom (2013).
*Abstract Algebra: An Inquiry Based Approach*. CRC Press. p. 425. ISBN 978-1-4665-6708-5. - W.R. Scott:
*Group Theory*, Prentice Hall, 1964, p. 27.