Correspondence theorem (group theory)
In the area of mathematics known as group theory, the correspondence theorem, sometimes referred to as the fourth isomorphism theorem or the lattice theorem, 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.
- 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.
- 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.
- 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.