Isomorphism theorems

Summary

In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.

History

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The isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, which was published in 1927 in Mathematische Annalen. Less general versions of these theorems can be found in work of Richard Dedekind and previous papers by Noether.

Three years later, B.L. van der Waerden published his influential Moderne Algebra, the first abstract algebra textbook that took the groups-rings-fields approach to the subject. Van der Waerden credited lectures by Noether on group theory and Emil Artin on algebra, as well as a seminar conducted by Artin, Wilhelm Blaschke, Otto Schreier, and van der Waerden himself on ideals as the main references. The three isomorphism theorems, called homomorphism theorem, and two laws of isomorphism when applied to groups, appear explicitly.

Groups

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We first present the isomorphism theorems of the groups.

Theorem A (groups)

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Diagram of the fundamental theorem on homomorphisms

Let G and H be groups, and let f : G → H be a homomorphism. Then:

  1. The kernel of f is a normal subgroup of G,
  2. The image of f is a subgroup of H, and
  3. The image of f is isomorphic to the quotient group G / ker(f).

In particular, if f is surjective then H is isomorphic to G / ker(f).

This theorem is usually called the first isomorphism theorem.

Theorem B (groups)

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Diagram for theorem B4. The two quotient groups (dotted) are isomorphic.

Let   be a group. Let   be a subgroup of  , and let   be a normal subgroup of  . Then the following hold:

  1. The product   is a subgroup of  ,
  2. The subgroup   is a normal subgroup of  ,
  3. The intersection   is a normal subgroup of  , and
  4. The quotient groups   and   are isomorphic.

Technically, it is not necessary for   to be a normal subgroup, as long as   is a subgroup of the normalizer of   in  . In this case,   is not a normal subgroup of  , but   is still a normal subgroup of the product  .

This theorem is sometimes called the second isomorphism theorem,[1] diamond theorem[2] or the parallelogram theorem.[3]

An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line starts with setting  , the group of invertible 2 × 2 complex matrices,  , the subgroup of determinant 1 matrices, and   the normal subgroup of scalar matrices  , we have  , where   is the identity matrix, and  . Then the second isomorphism theorem states that:

 

Theorem C (groups)

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Let   be a group, and   a normal subgroup of  . Then

  1. If   is a subgroup of   such that  , then   has a subgroup isomorphic to  .
  2. Every subgroup of   is of the form   for some subgroup   of   such that  .
  3. If   is a normal subgroup of   such that  , then   has a normal subgroup isomorphic to  .
  4. Every normal subgroup of   is of the form   for some normal subgroup   of   such that  .
  5. If   is a normal subgroup of   such that  , then the quotient group   is isomorphic to  .

The last statement is sometimes referred to as the third isomorphism theorem. The first four statements are often subsumed under Theorem D below, and referred to as the lattice theorem, correspondence theorem, or fourth isomorphism theorem.

Theorem D (groups)

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Let   be a group, and   a normal subgroup of  . The canonical projection homomorphism   defines a bijective correspondence between the set of subgroups of   containing   and the set of (all) subgroups of  . Under this correspondence normal subgroups correspond to normal subgroups.

This theorem is sometimes called the correspondence theorem, the lattice theorem, and the fourth isomorphism theorem.

The Zassenhaus lemma (also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem.[4]

Discussion

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The first isomorphism theorem can be expressed in category theoretical language by saying that the category of groups is (normal epi, mono)-factorizable; in other words, the normal epimorphisms and the monomorphisms form a factorization system for the category. This is captured in the commutative diagram in the margin, which shows the objects and morphisms whose existence can be deduced from the morphism  . The diagram shows that every morphism in the category of groups has a kernel in the category theoretical sense; the arbitrary morphism f factors into  , where ι is a monomorphism and π is an epimorphism (in a conormal category, all epimorphisms are normal). This is represented in the diagram by an object   and a monomorphism   (kernels are always monomorphisms), which complete the short exact sequence running from the lower left to the upper right of the diagram. The use of the exact sequence convention saves us from having to draw the zero morphisms from   to   and  .

If the sequence is right split (i.e., there is a morphism σ that maps   to a π-preimage of itself), then G is the semidirect product of the normal subgroup   and the subgroup  . If it is left split (i.e., there exists some   such that  ), then it must also be right split, and   is a direct product decomposition of G. In general, the existence of a right split does not imply the existence of a left split; but in an abelian category (such as that of abelian groups), left splits and right splits are equivalent by the splitting lemma, and a right split is sufficient to produce a direct sum decomposition  . In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence  .

In the second isomorphism theorem, the product SN is the join of S and N in the lattice of subgroups of G, while the intersection S ∩ N is the meet.

The third isomorphism theorem is generalized by the nine lemma to abelian categories and more general maps between objects.

Note on numbers and names

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Below we present four theorems, labelled A, B, C and D. They are often numbered as "First isomorphism theorem", "Second..." and so on; however, there is no universal agreement on the numbering. Here we give some examples of the group isomorphism theorems in the literature. Notice that these theorems have analogs for rings and modules.

Comparison of the names of the group isomorphism theorems
Comment Author Theorem A Theorem B Theorem C
No "third" theorem Jacobson[5] Fundamental theorem of homomorphisms (Second isomorphism theorem) "often called the first isomorphism theorem"
van der Waerden,[6] Durbin[8] Fundamental theorem of homomorphisms First isomorphism theorem Second isomorphism theorem
Knapp[9] (No name) Second isomorphism theorem First isomorphism theorem
Grillet[10] Homomorphism theorem Second isomorphism theorem First isomorphism theorem
Three numbered theorems (Other convention per Grillet) First isomorphism theorem Third isomorphism theorem Second isomorphism theorem
Rotman[11] First isomorphism theorem Second isomorphism theorem Third isomorphism theorem
Fraleigh[12] Fundamental homomorphism theorem or first isomorphism theorem Second isomorphism theorem Third isomorphism theorem
Dummit & Foote[13] First isomorphism theorem Second or Diamond isomorphism theorem Third isomorphism theorem
No numbering Milne[1] Homomorphism theorem Isomorphism theorem Correspondence theorem
Scott[14] Homomorphism theorem Isomorphism theorem Freshman theorem

It is less common to include the Theorem D, usually known as the lattice theorem or the correspondence theorem, as one of isomorphism theorems, but when included, it is the last one.

Rings

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The statements of the theorems for rings are similar, with the notion of a normal subgroup replaced by the notion of an ideal.

Theorem A (rings)

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Let   and   be rings, and let   be a ring homomorphism. Then:

  1. The kernel of   is an ideal of  ,
  2. The image of   is a subring of  , and
  3. The image of   is isomorphic to the quotient ring  .

In particular, if   is surjective then   is isomorphic to  .[15]

Theorem B (rings)

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Let R be a ring. Let S be a subring of R, and let I be an ideal of R. Then:

  1. The sum S + I = {s + i | s ∈ Si ∈ I } is a subring of R,
  2. The intersection S ∩ I is an ideal of S, and
  3. The quotient rings (S + I) / I and S / (S ∩ I) are isomorphic.

Theorem C (rings)

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Let R be a ring, and I an ideal of R. Then

  1. If   is a subring of   such that  , then   is a subring of  .
  2. Every subring of   is of the form   for some subring   of   such that  .
  3. If   is an ideal of   such that  , then   is an ideal of  .
  4. Every ideal of   is of the form   for some ideal   of   such that  .
  5. If   is an ideal of   such that  , then the quotient ring   is isomorphic to  .

Theorem D (rings)

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Let   be an ideal of  . The correspondence   is an inclusion-preserving bijection between the set of subrings   of   that contain   and the set of subrings of  . Furthermore,   (a subring containing  ) is an ideal of   if and only if   is an ideal of  .[16]

Modules

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The statements of the isomorphism theorems for modules are particularly simple, since it is possible to form a quotient module from any submodule. The isomorphism theorems for vector spaces (modules over a field) and abelian groups (modules over  ) are special cases of these. For finite-dimensional vector spaces, all of these theorems follow from the rank–nullity theorem.

In the following, "module" will mean "R-module" for some fixed ring R.

Theorem A (modules)

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Let M and N be modules, and let φ : M → N be a module homomorphism. Then:

  1. The kernel of φ is a submodule of M,
  2. The image of φ is a submodule of N, and
  3. The image of φ is isomorphic to the quotient module M / ker(φ).

In particular, if φ is surjective then N is isomorphic to M / ker(φ).

Theorem B (modules)

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Let M be a module, and let S and T be submodules of M. Then:

  1. The sum S + T = {s + t | s ∈ St ∈ T} is a submodule of M,
  2. The intersection S ∩ T is a submodule of M, and
  3. The quotient modules (S + T) / T and S / (S ∩ T) are isomorphic.

Theorem C (modules)

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Let M be a module, T a submodule of M.

  1. If   is a submodule of   such that  , then   is a submodule of  .
  2. Every submodule of   is of the form   for some submodule   of   such that  .
  3. If   is a submodule of   such that  , then the quotient module   is isomorphic to  .

Theorem D (modules)

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Let   be a module,   a submodule of  . There is a bijection between the submodules of   that contain   and the submodules of  . The correspondence is given by   for all  . This correspondence commutes with the processes of taking sums and intersections (i.e., is a lattice isomorphism between the lattice of submodules of   and the lattice of submodules of   that contain  ).[17]

Universal algebra

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To generalise this to universal algebra, normal subgroups need to be replaced by congruence relations.

A congruence on an algebra   is an equivalence relation   that forms a subalgebra of   considered as an algebra with componentwise operations. One can make the set of equivalence classes   into an algebra of the same type by defining the operations via representatives; this will be well-defined since   is a subalgebra of  . The resulting structure is the quotient algebra.

Theorem A (universal algebra)

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Let   be an algebra homomorphism. Then the image of   is a subalgebra of  , the relation given by   (i.e. the kernel of  ) is a congruence on  , and the algebras   and   are isomorphic. (Note that in the case of a group,   iff  , so one recovers the notion of kernel used in group theory in this case.)

Theorem B (universal algebra)

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Given an algebra  , a subalgebra   of  , and a congruence   on  , let   be the trace of   in   and   the collection of equivalence classes that intersect  . Then

  1.   is a congruence on  ,
  2.   is a subalgebra of  , and
  3. the algebra   is isomorphic to the algebra  .

Theorem C (universal algebra)

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Let   be an algebra and   two congruence relations on   such that  . Then   is a congruence on  , and   is isomorphic to  

Theorem D (universal algebra)

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Let   be an algebra and denote   the set of all congruences on  . The set   is a complete lattice ordered by inclusion.[18] If   is a congruence and we denote by   the set of all congruences that contain   (i.e.   is a principal filter in  , moreover it is a sublattice), then the map   is a lattice isomorphism.[19][20]

Notes

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  1. ^ a b Milne (2013), Chap. 1, sec. Theorems concerning homomorphisms
  2. ^ I. Martin Isaacs (1994). Algebra: A Graduate Course. American Mathematical Soc. p. 33. ISBN 978-0-8218-4799-2.
  3. ^ Paul Moritz Cohn (2000). Classic Algebra. Wiley. p. 245. ISBN 978-0-471-87731-8.
  4. ^ Wilson, Robert A. (2009). The Finite Simple Groups. Graduate Texts in Mathematics 251. Vol. 251. Springer-Verlag London. p. 7. doi:10.1007/978-1-84800-988-2. ISBN 978-1-4471-2527-3.
  5. ^ Jacobson (2009), sec 1.10
  6. ^ van der Waerden, Algebra (1994).
  7. ^ Durbin (2009), sec. 54
  8. ^ [the names are] essentially the same as [van der Waerden 1994][7]
  9. ^ Knapp (2016), sec IV 2
  10. ^ Grillet (2007), sec. I 5
  11. ^ Rotman (2003), sec. 2.6
  12. ^ Fraleigh (2003), Chap. 14, 34
  13. ^ Dummit, David Steven (2004). Abstract algebra. Richard M. Foote (Third ed.). Hoboken, NJ. pp. 97–98. ISBN 0-471-43334-9. OCLC 52559229.{{cite book}}: CS1 maint: location missing publisher (link)
  14. ^ Scott (1964), secs 2.2 and 2.3
  15. ^ Moy, Samuel (2022). "An Introduction to the Theory of Field Extensions" (PDF). UChicago Department of Math. Retrieved Dec 20, 2022.
  16. ^ Dummit, David S.; Foote, Richard M. (2004). Abstract algebra. Hoboken, NJ: Wiley. p. 246. ISBN 978-0-471-43334-7.
  17. ^ Dummit and Foote (2004), p. 349
  18. ^ Burris and Sankappanavar (2012), p. 37
  19. ^ Burris and Sankappanavar (2012), p. 49
  20. ^ Sun, William. "Is there a general form of the correspondence theorem?". Mathematics StackExchange. Retrieved 20 July 2019.

References

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  • Noether, Emmy, Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, Mathematische Annalen 96 (1927) pp. 26–61
  • McLarty, Colin, "Emmy Noether's 'Set Theoretic' Topology: From Dedekind to the rise of functors". The Architecture of Modern Mathematics: Essays in history and philosophy (edited by Jeremy Gray and José Ferreirós), Oxford University Press (2006) pp. 211–35.
  • Jacobson, Nathan (2009), Basic algebra, vol. 1 (2nd ed.), Dover, ISBN 9780486471891
  • Cohn, Paul M., Universal algebra, Chapter II.3 p. 57
  • Milne, James S. (2013), Group Theory, 3.13
  • van der Waerden, B. I. (1994), Algebra, vol. 1 (9 ed.), Springer-Verlag
  • Dummit, David S.; Foote, Richard M. (2004). Abstract algebra. Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7.
  • Burris, Stanley; Sankappanavar, H. P. (2012). A Course in Universal Algebra (PDF). ISBN 978-0-9880552-0-9.
  • Scott, W. R. (1964), Group Theory, Prentice Hall
  • Durbin, John R. (2009). Modern Algebra: An Introduction (6 ed.). Wiley. ISBN 978-0-470-38443-5.
  • Knapp, Anthony W. (2016), Basic Algebra (Digital second ed.)
  • Grillet, Pierre Antoine (2007), Abstract Algebra (2 ed.), Springer
  • Rotman, Joseph J. (2003), Advanced Modern Algebra (2 ed.), Prentice Hall, ISBN 0130878685
  • Hungerford, Thomas W. (1980), Algebra (Graduate Texts in Mathematics, 73), Springer, ISBN 0387905189