Correspondence theorem

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In 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 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 . Loosely speaking, 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,
, a normal subgroup of G,
, the set of all subgroups A of G that contain N, and
, 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 then

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. A proof of the correspondence theorem can be found here.

Similar results hold for rings, modules, vector spaces, and algebras. More generally an analogous result that concerns congruence relations instead of normal subgroups holds for any algebraic structure.

See also

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References

  1. Derek John Scott Robinson (2003). An Introduction to Abstract Algebra . Walter de Gruyter. p.  64. ISBN   978-3-11-017544-8.
  2. J. F. Humphreys (1996). A Course in Group Theory . Oxford University Press. p.  65. ISBN   978-0-19-853459-4.
  3. H.E. Rose (2009). A Course on Finite Groups . Springer. p.  78. ISBN   978-1-84882-889-6.
  4. J.L. Alperin; Rowen B. Bell (1995). Groups and Representations . Springer. p.  11. ISBN   978-1-4612-0799-3.
  5. I. Martin Isaacs (1994). Algebra: A Graduate Course . American Mathematical Soc. p.  35. ISBN   978-0-8218-4799-2.
  6. 1 2 Joseph Rotman (1995). An Introduction to the Theory of Groups (4th ed.). Springer. pp.  37–38. ISBN   978-1-4612-4176-8.
  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.