# Isomorphism theorems

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

## History[edit]

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.

## Groups[edit]

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* and Pinter's *A Book of Abstract 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 theorems[edit]

#### First isomorphism theorem[edit]

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

- The kernel of
*φ*is a normal subgroup of*G*, - The image of
*φ*is a subgroup of*H*, and - 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 theorem[edit]

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

- The product is a subgroup of ,
- The intersection is a normal subgroup of , and
- 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 theorem[edit]

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

- If is a subgroup of such that , then is a subgroup of .
- Every subgroup of is of the form , for some subgroup of such that .
- If is a normal subgroup of such that , then is a normal subgroup of .
- Every normal subgroup of is of the form , for some normal subgroup of such that .
- 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]}

#### Fourth isomorphism theorem[edit]

It can refer either to correspondence theorem^{[7]} or to butterfly lemma.

### Discussion[edit]

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.

## Rings[edit]

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 theorem[edit]

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

- The kernel of
*φ*is an ideal of*R*, - The image of
*φ*is a subring of*S*, and - 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 theorem[edit]

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

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

### Third isomorphism theorem[edit]

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

- If is a subring of such that , then is a subring of .
- Every subring of is of the form , for some subring of such that .
- If is an ideal of such that , then is an ideal of .
- Every ideal of is of the form , for some ideal of such that .
- If is an ideal of such that , then the quotient ring is isomorphic to .

### Fourth isomorphism theorem[edit]

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 .^{[8]}

## Modules[edit]

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 theorem[edit]

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

- The kernel of
*φ*is a submodule of*M*, - The image of
*φ*is a submodule of*N*, and - 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 theorem[edit]

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

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

### Third isomorphism theorem[edit]

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

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

### Fourth isomorphism theorem[edit]

Let be a module, its submodule. There is a bijection between the submodules of which 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 which contain ).^{[9]}

## General[edit]

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 theorem[edit]

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 theorem[edit]

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

- is a congruence on ,
- is a subalgebra of , and
- the algebra is isomorphic to the algebra .

### Third isomorphism theorem[edit]

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

### Fourth isomorphism theorem[edit]

Let be an algebra and denote the set of all congruences on . The set
is a complete lattice ordered by inclusion.^{[10]}
If is a congruence and we denote by the set of all congruences that contain (i.e. is a principal (order) ideal in , moreover it is a sublattice), then
the map is a lattice isomorphism.^{[11]}^{[12]}

## See also[edit]

- Butterfly lemma, sometimes called the fourth isomorphism theorem
- Correspondence theorem, also sometimes called the fourth isomorphism theorem

## Notes[edit]

**^**Jerry Shurman Group isomorphism theorems from Reed College**^**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*.**^**John R. Durbin (2009).*Modern Algebra: An Introduction*. Wiley. p. 238. ISBN 978-0-470-38443-5.**^**I. Martin Isaacs (1994).*Algebra: A Graduate Course*. American Mathematical Soc. p. 33. ISBN 978-0-8218-4799-2.**^**Paul Moritz Cohn (2000).*Classic Algebra*. Wiley. p. 245. ISBN 978-0-471-87731-8.**^**W.R. Scott:*Group Theory*, Prentice Hall, 1964, p. 33**^**Dummit, David S.; Foote, Richard M. (2004).*Abstract algebra*. Hoboken, NJ: Wiley. p. 99. ISBN 978-0-471-43334-7.**^**Dummit, David S.; Foote, Richard M. (2004).*Abstract algebra*. Hoboken, NJ: Wiley. p. 246. ISBN 978-0-471-43334-7.**^**Dummit, David S.; Foote, Richard M. (2004).*Abstract algebra*. Hoboken, NJ: Wiley. p. 349. ISBN 978-0-471-43334-7.**^**Burris, Stanley; Sankappanavar, H. P. (2012).*A Course in Universal Algebra*(PDF). p. 37. ISBN 978-0-9880552-0-9.**^**Burris, Stanley; Sankappanavar, H. P. (2012).*A Course in Universal Algebra*(PDF). p. 49. ISBN 978-0-9880552-0-9.**^**William Sun, (https://math.stackexchange.com/users/413924/william-sun). "Is there a general form of the correspondence theorem?".*Mathematics StackExchange*. Retrieved 20 July 2019.

## References[edit]

- 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