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In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are three 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.

Contents

HistoryEdit

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 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.

GroupsEdit

The first instance of the isomorphism theorems that we present occurs in the category of abstract groups.[1] Note that some sources switch the numbering of the second and third theorems.[2] Another variation encountered in the literature, particularly in Van der Waerden's Algebra, is to call first isomorphism theorem the Fundamental Homomorphism Theorem and consequently to decrement the numbering of the remaining isomorphism theorems by one.[3] Finally, in the most extensive numbering scheme, the lattice theorem (also known as the correspondence theorem) is sometimes referred to as the fourth isomorphism theorem.

Statement of the theoremsEdit

First isomorphism theoremEdit

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

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

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

Second isomorphism theoremEdit

 
Diagram for the Second Isomorphism Theorem

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 intersection   is a normal subgroup of  , and
  3. 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, the intersection   is not a normal subgroup of  , but it is still a normal subgroup of  .

This isomorphism theorem has been called the "diamond theorem" due to the shape of the resulting subgroup lattice with   at the top,   at the bottom and with   and   to the sides.[4] It has even been called the "parallelogram theorem" because in the resulting subgroup lattice the two sides assumed to represent the quotient groups   and   are "equal" in the sense of isomorphism.[5]

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 2x2 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:

 

Third isomorphism theoremEdit

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

  1. If   is a subgroup of   such that  , then   is a subgroup of  .
  2. Every subgroup of   is of the form  , for some subgroup   of   such that  .
  3. If   is a normal subgroup of   such that  , then   is a normal subgroup of  .
  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  .

W.R. Scott calls it the "Freshman theorem" because the result simply follows by "cancellation" of  , as one would naively assume from cancellation of numerical fractions.[6]

DiscussionEdit

First isomorphism theorem
 

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 the 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.

RingsEdit

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

First isomorphism theoremEdit

Let R and S be rings, and let φR → S be a ring homomorphism. Then:

  1. The kernel of φ is an ideal of R,
  2. The image of φ is a subring of S, and
  3. The image of φ is isomorphic to the quotient ring R / ker(φ).

In particular, if φ is surjective then S is isomorphic to R / ker(φ).

Second isomorphism theoremEdit

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.

Third isomorphism theoremEdit

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  .

ModulesEdit

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.

First isomorphism theoremEdit

Let M and N be modules, and let φM → 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).

Second isomorphism theoremEdit

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.

Third isomorphism theoremEdit

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  .

GeneralEdit

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

A congruence on an algebra   is an equivalence relation   which 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.

First isomorphism theoremEdit

Let   be an algebra homomorphism. Then the image of   is a subalgebra of  , the relation given by   (i.e. the kernel of f) is a congruence on  , and the algebras   and   are isomorphic. (Note that in the case of a group, f(x) = f(y) iff f(xy−1) = 1, so one recovers the notion of kernel used in group theory in this case.)

Second isomorphism theoremEdit

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 (i)   is a congruence on  , (ii)   is a subalgebra of  , and (iii) the algebra   is isomorphic to the algebra  .

Third isomorphism theoremEdit

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

See alsoEdit

NotesEdit

  1. ^ Jerry Shurman Group isomorphism theorems from Reed College
  2. ^ Jacobson (2009), p. 101, use "first" for the isomorphism of the modules (S + T) / T and S / (S ∩ T), and "second" for (M / T) / (S / T) and M / S.
  3. ^ John R. Durbin (2009). Modern Algebra: An Introduction. Wiley. p. 238. ISBN 978-0-470-38443-5.
  4. ^ I. Martin Isaacs (1994). Algebra: A Graduate Course. American Mathematical Soc. p. 33. ISBN 978-0-8218-4799-2.
  5. ^ Paul Moritz Cohn (2000). Classic Algebra. Wiley. p. 245. ISBN 978-0-471-87731-8.
  6. ^ W.R. Scott: Group Theory, Prentice Hall, 1964, p. 33

ReferencesEdit

  • Emmy Noether, Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, Mathematische Annalen 96 (1927) pp. 26–61
  • Colin McLarty, 'Emmy Noether’s ‘Set Theoretic’ Topology: From Dedekind to the rise of functors' in 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, 2 (2nd ed.), Dover, ISBN 978-0-486-47187-7
  • Paul M. Cohn, Universal algebra, Chapter II.3 p. 57

External linksEdit