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] 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[edit]

Notes[edit]

  1. ^ 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. 
  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[edit]

  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. ^ a b 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. ^ Jonathan K. Hodge; Steven Schlicker; Ted Sundstrom (2013). Abstract Algebra: An Inquiry Based Approach. CRC Press. p. 425. ISBN 978-1-4665-6708-5. 
  10. ^ W.R. Scott: Group Theory, Prentice Hall, 1964, p. 27.