Editing Correspondence theorem

{{Short description|Theorem in group theory}}
In [[group theory]], the '''correspondence theorem'''<ref name="Robinson2003">
{{cite book
|author=Derek John Scott Robinson
|title=An Introduction to Abstract Algebra|url=https://archive.org/details/introductiontoab00robi_926
|url-access=limited
|year=2003
|publisher=Walter de Gruyter
|isbn=978-3-11-017544-8
|page=[https://archive.org/details/introductiontoab00robi_926/page/n73 64]
}}</ref><ref name="Humphreys1996">
{{cite book
|author=J. F. Humphreys
|title=A Course in Group Theory
|url=https://archive.org/details/coursegrouptheor00hump
|url-access=limited
|year=1996
|publisher=Oxford University Press
|isbn=978-0-19-853459-4
|page=[https://archive.org/details/coursegrouptheor00hump/page/n77 65]
}}</ref><ref name="Rose2009">
{{cite book
|author=H.E. Rose
|title=A Course on Finite Groups|url=https://archive.org/details/courseonfinitegr00rose_817
|url-access=limited
|year=2009
|publisher=Springer
|isbn=978-1-84882-889-6
|page=[https://archive.org/details/courseonfinitegr00rose_817/page/n89 78]
}}</ref><ref name="AlperinBell1995">
{{cite book
|author1=J.L. Alperin
|author2=Rowen B. Bell
|title=Groups and Representations
|url=https://archive.org/details/groupsrepresenta00alpe_213
|url-access=limited
|year=1995
|publisher=Springer
|isbn=978-1-4612-0799-3
|page=[https://archive.org/details/groupsrepresenta00alpe_213/page/n65 11]
}}</ref><ref name="Isaacs1994">{{cite book
|author=I. Martin Isaacs
|title=Algebra: A Graduate Course
|url=https://archive.org/details/algebragraduatec00isaa
|url-access=limited
|year=1994
|publisher=American Mathematical Soc.
|isbn=978-0-8218-4799-2
|page=[https://archive.org/details/algebragraduatec00isaa/page/n47 35]
}}</ref><ref name="Rotman1995">
{{cite book
|author=Joseph Rotman
|title=An Introduction to the Theory of Groups
|url=https://archive.org/details/introductiontoth00rotm_373
|url-access=limited
|year=1995
|publisher=Springer
|isbn=978-1-4612-4176-8
|pages=[https://archive.org/details/introductiontoth00rotm_373/page/n57 37]–38
|edition=4th
}}</ref><ref name="Nicholson2012">
{{cite book
|author=W. Keith Nicholson
|title=Introduction to Abstract Algebra
|year=2012
|publisher=John Wiley & Sons
|isbn=978-1-118-31173-8
|page=352
|edition=4th
}}</ref><ref name="Roman2011">
{{cite book
|author=Steven Roman
|title=Fundamentals of Group Theory: An Advanced Approach
|year=2011
|publisher=Springer Science & Business Media
|isbn=978-0-8176-8301-6
|pages=113–115
}}</ref> (also the '''lattice theorem''',<ref>W.R. Scott: ''Group Theory'', Prentice Hall, 1964, p. 27.</ref> and [[isomorphism theorems#Note on numbers and names|variously and ambiguously the '''third''' and '''fourth isomorphism theorem''']]<ref name="Rotman1995"/><ref name="HodgeSchlicker2013">
{{cite book
|author1=Jonathan K. Hodge
|author2=Steven Schlicker
|author3=Ted Sundstrom
|title=Abstract Algebra: An Inquiry Based Approach
|year=2013
|publisher=CRC Press
|isbn=978-1-4665-6708-5
|page=425
}}</ref>) states that if <math>N</math> is a [[normal subgroup]] of a [[group (mathematics)|group]] <math>G</math>, then there exists a [[bijection]] from the set of all [[subgroups]] <math>A</math> of <math>G</math> containing <math>N</math>, onto the set of all subgroups of the [[quotient group]] <math>G/N</math>.  Loosely speaking, the structure of the subgroups of <math>G/N</math> is exactly the same as the structure of the subgroups of <math>G</math> containing <math>N</math>, with <math>N</math> collapsed to the [[identity element]].

Specifically, if 
: <math>G</math> is a group,
: <math>N \triangleleft G</math>, a [[normal subgroup]] of <math>G</math>,
: <math>\mathcal{G} = \{ A \mid  N \subseteq A \leq G \}</math>, the set of all subgroups <math>A</math> of <math>G</math> that contain <math>N</math>, and
: <math>\mathcal{N} = \{ S \mid S \leq G/N \}</math>, the set of all subgroups of <math>G/N</math>,

then there is a bijective map <math>\phi: \mathcal{G} \to \mathcal{N}</math> such that
: <math>\phi(A) = A/N</math> for all <math>A \in \mathcal{G}.</math>

One further has that if <math>A</math> and <math>B</math> are in <math>\mathcal{G}</math> then
* <math>A \subseteq B</math> if and only if <math>A/N \subseteq B/N</math>;
* if <math>A \subseteq B</math> then <math>|B:A| = |B/N:A/N|</math>, where <math>|B:A|</math> is the [[index (group theory)|index]] of <math>A</math> in <math>B</math> (the number of [[coset]]s <math>bA</math> of <math>A</math> in <math>B</math>);
* <math>\langle A,B \rangle / N = \left\langle A/N, B/N \right\rangle,</math> where <math>\langle A, B \rangle</math> is the subgroup of <math>G</math> [[Generating set of a group|generated]] by  <math>A\cup B;</math>
* <math>(A \cap B)/N = A/N \cap B/N</math>, and 
* <math>A</math> is a normal subgroup of <math>G</math> if and only if <math>A/N</math> is a normal subgroup of <math>G/N</math>.

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 [[Galois connection|monotone Galois connection]] <math>(f^*, f_*)</math> between the [[lattice of subgroups]] of <math>G</math> (not necessarily containing <math>N</math>) and the lattice of subgroups of <math>G/N</math>: the lower adjoint of a subgroup <math>H</math> of <math>G</math> is given by <math>f^*(H) = HN/N</math> and the upper adjoint of a subgroup <math>K/N</math> of <math>G/N</math> is a given by <math>f_*(K/N) = K</math>. The associated [[closure operator]] on subgroups of <math>G</math> is <math>\bar H = HN</math>; the associated [[kernel operator]] on subgroups of <math>G/N</math> is the identity. A proof of the correspondence theorem can be found [https://proofwiki.org/wiki/Correspondence_Theorem_(Group_Theory) here].

<!-- This needs to be expanded to include rings, etc. -->
Similar results hold for [[ring (mathematics)|rings]], [[module (mathematics)|modules]], [[vector space]]s, and [[algebra over a field|algebras]]. More generally an [[Isomorphism_theorems#Theorem_D_(universal_algebra)|analogous result]] that concerns [[congruence relation]]s instead of normal subgroups holds for any [[algebraic structure]].

== See also ==
* [[Modular lattice]]

==References==
{{reflist}}

[[Category:Isomorphism theorems]]