Correspondence theorem (group theory)

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In the area of mathematics known as group theory, the correspondence theorem[1][2][3][4][5][6][7][8] (also the lattice theorem,[9] and variously and ambiguously the third and fourth isomorphism theorem[6][10]) states that if [math]\displaystyle{ N }[/math] is a normal subgroup of a group [math]\displaystyle{ G }[/math], then there exists a bijection from the set of all subgroups [math]\displaystyle{ A }[/math] of [math]\displaystyle{ G }[/math] containing [math]\displaystyle{ N }[/math], onto the set of all subgroups of the quotient group [math]\displaystyle{ G/N }[/math]. The structure of the subgroups of [math]\displaystyle{ G/N }[/math] is exactly the same as the structure of the subgroups of [math]\displaystyle{ G }[/math] containing [math]\displaystyle{ N }[/math], with [math]\displaystyle{ N }[/math] collapsed to the identity element.

Specifically, if

G is a group,
[math]\displaystyle{ N \triangleleft G }[/math], a normal subgroup of G,
[math]\displaystyle{ \mathcal{G} = \{ \forall A \mid N \subseteq A \lt G \} }[/math], the set of all subgroups of G such that each subgroup A contains N, and
[math]\displaystyle{ \mathcal{N} = \{ \forall S \mid S \lt G/N \} }[/math], the set of all subgroups of G/N,

then there is a bijective map [math]\displaystyle{ \phi: \mathcal{G} \to \mathcal{N} }[/math] such that

[math]\displaystyle{ \phi(A) = A/N }[/math] for all [math]\displaystyle{ A \in \mathcal{G}. }[/math]

One further has that if A and B are in [math]\displaystyle{ \mathcal{G} }[/math] then

  • [math]\displaystyle{ A \subseteq B }[/math] if and only if [math]\displaystyle{ A/N \subseteq B/N }[/math];
  • if [math]\displaystyle{ A \subseteq B }[/math] then [math]\displaystyle{ |B:A| = |B/N:A/N| }[/math], where [math]\displaystyle{ |B:A| }[/math] is the index of A in B (the number of cosets bA of A in B);
  • [math]\displaystyle{ \langle A,B \rangle / N = \left\langle A/N, B/N \right\rangle, }[/math] where [math]\displaystyle{ \langle A, B \rangle }[/math] is the subgroup of [math]\displaystyle{ G }[/math] generated by [math]\displaystyle{ A\cup B; }[/math]
  • [math]\displaystyle{ (A \cap B)/N = A/N \cap B/N }[/math], and
  • [math]\displaystyle{ A }[/math] is a normal subgroup of [math]\displaystyle{ G }[/math] if and only if [math]\displaystyle{ A/N }[/math] is a normal subgroup of [math]\displaystyle{ 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 monotone Galois connection [math]\displaystyle{ (f^*, f_*) }[/math] between the lattice of subgroups of [math]\displaystyle{ G }[/math] (not necessarily containing [math]\displaystyle{ N }[/math]) and the lattice of subgroups of [math]\displaystyle{ G/N }[/math]: the lower adjoint of a subgroup [math]\displaystyle{ H }[/math] of [math]\displaystyle{ G }[/math] is given by [math]\displaystyle{ f^*(H) = HN/N }[/math] and the upper adjoint of a subgroup [math]\displaystyle{ K/N }[/math] of [math]\displaystyle{ G/N }[/math] is a given by [math]\displaystyle{ f_*(K/N) = K }[/math]. The associated closure operator on subgroups of [math]\displaystyle{ G }[/math] is [math]\displaystyle{ \bar H = HN }[/math]; the associated kernel operator on subgroups of [math]\displaystyle{ G/N }[/math] is the identity. A proof of the correspondence theorem can be found here.

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

See also

References

  1. ↑ Derek John Scott Robinson (2003). An Introduction to Abstract Algebra. Walter de Gruyter. p. 64. ISBN 978-3-11-017544-8. https://archive.org/details/introductiontoab00robi_926. 
  2. ↑ J. F. Humphreys (1996). A Course in Group Theory. Oxford University Press. p. 65. ISBN 978-0-19-853459-4. https://archive.org/details/coursegrouptheor00hump. 
  3. ↑ H.E. Rose (2009). A Course on Finite Groups. Springer. p. 78. ISBN 978-1-84882-889-6. https://archive.org/details/courseonfinitegr00rose_817. 
  4. ↑ J.L. Alperin; Rowen B. Bell (1995). Groups and Representations. Springer. p. 11. ISBN 978-1-4612-0799-3. https://archive.org/details/groupsrepresenta00alpe_213. 
  5. ↑ I. Martin Isaacs (1994). Algebra: A Graduate Course. American Mathematical Soc.. p. 35. ISBN 978-0-8218-4799-2. https://archive.org/details/algebragraduatec00isaa. 
  6. ↑ 6.0 6.1 Joseph Rotman (1995). An Introduction to the Theory of Groups (4th ed.). Springer. pp. 37–38. ISBN 978-1-4612-4176-8. https://archive.org/details/introductiontoth00rotm_373. 
  7. ↑ W. Keith Nicholson (2012). Introduction to Abstract Algebra (4th ed.). John Wiley & Sons. p. 352. ISBN 978-1-118-31173-8. 
  8. ↑ Steven Roman (2011). Fundamentals of Group Theory: An Advanced Approach. Springer Science & Business Media. pp. 113–115. ISBN 978-0-8176-8301-6. 
  9. ↑ W.R. Scott: Group Theory, Prentice Hall, 1964, p. 27.
  10. ↑ Jonathan K. Hodge; Steven Schlicker; Ted Sundstrom (2013). Abstract Algebra: An Inquiry Based Approach. CRC Press. p. 425. ISBN 978-1-4665-6708-5. 




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