# Isomorphism theorems

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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

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

We first present the isomorphism theorems of the groups.

### Note on numbers and names

Below we present 4 theorems, calling A, B, C and D. They are often numbered as "First isomorphism theorem", "Second..." and so on, but the numbering convetion is completely lacking. Here we give some random examples in the literature:

Theorem A Theorem B Theorem C Jacobson[1] van der Waerden[2], Durbin[4] No "third" theorem Fundamental theorem of homomorphisms (second isomorphism theorem) "often called the first isomorphism theorem" Fundamental theorem of homomorphisms first isomorphism theorem second isomorphism theorem (no name) Second isomorphism theorem First isomorphism theorem Homomorphism theorem Second isomorphism theorem First isomorphism theorem Three numbered theorems Other convention mentioned in Grillet First isomorphism theorem Third isomorphism theorem Second isomorphism theorem First isomorphism theorem Second isomorphism theorem Third isomorphism theorem No numbering Homomorphism theorem Isomorphism theorem Correspondence theorem 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", to one of isomorphism theorems, but when they do, it is the last one.

### Statement of the theorems

#### Theorem A

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).

#### Theorem B

Diagram for the theorem B

Let ${\displaystyle G}$  be a group. Let ${\displaystyle S}$  be a subgroup of ${\displaystyle G}$ , and let ${\displaystyle N}$  be a normal subgroup of ${\displaystyle G}$ . Then the following hold:

1. The product ${\displaystyle SN}$  is a subgroup of ${\displaystyle G}$ ,
2. The intersection ${\displaystyle S\cap N}$  is a normal subgroup of ${\displaystyle S}$ , and
3. The quotient groups ${\displaystyle (SN)/N}$  and ${\displaystyle S/(S\cap N)}$  are isomorphic.

Technically, it is not necessary for ${\displaystyle N}$  to be a normal subgroup, as long as ${\displaystyle S}$  is a subgroup of the normalizer of ${\displaystyle N}$  in ${\displaystyle G}$ . In this case, the intersection ${\displaystyle S\cap N}$  is not a normal subgroup of ${\displaystyle G}$ , but it is still a normal subgroup of ${\displaystyle S}$ .

This theorem is sometimes called the "isomorphism theorem"[8], "diamond theorem"[10] or the "parallelogram theorem"[11].

An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line starts with setting ${\displaystyle G=GL_{2}(\mathbb {C} )}$ , the group of invertible 2x2 complex matrices, ${\displaystyle S=SL_{2}(\mathbb {C} )}$ , the subgroup of determinant 1 matrices, and ${\displaystyle N}$  the normal subgroup of scalar matrices ${\displaystyle \mathbb {C} ^{\times }\!I=\left\{{\bigl (}{\begin{smallmatrix}a&0\\0&a\end{smallmatrix}}{\bigr )}:a\in \mathbb {C} ^{\times }\right\}}$ , we have ${\displaystyle S\cap N=\{\pm I\}}$ , where ${\displaystyle I}$  is the identity matrix, and ${\displaystyle SN=GL_{2}(\mathbb {C} )}$ . Then the second isomorphism theorem states that:

${\displaystyle PGL_{2}(\mathbb {C} ):=GL_{2}(\mathbb {C} )/(\mathbb {C} ^{\times }\!I)\cong SL_{2}(\mathbb {C} )/\{\pm I\}=:PSL_{2}(\mathbb {C} )}$

#### Theorem C

Let ${\displaystyle G}$  be a group, and ${\displaystyle N}$  a normal subgroup of ${\displaystyle G}$ . Then

1. If ${\displaystyle K}$  is a subgroup of ${\displaystyle G}$  such that ${\displaystyle N\subseteq K\subseteq G}$ , then ${\displaystyle K/N}$  is a subgroup of ${\displaystyle G/N}$ .
2. Every subgroup of ${\displaystyle G/N}$  is of the form ${\displaystyle K/N}$ , for some subgroup ${\displaystyle K}$  of ${\displaystyle G}$  such that ${\displaystyle N\subseteq K\subseteq G}$ .
3. If ${\displaystyle K}$  is a normal subgroup of ${\displaystyle G}$  such that ${\displaystyle N\subseteq K\subseteq G}$ , then ${\displaystyle K/N}$  is a normal subgroup of ${\displaystyle G/N}$ .
4. Every normal subgroup of ${\displaystyle G/N}$  is of the form ${\displaystyle K/N}$ , for some normal subgroup ${\displaystyle K}$  of ${\displaystyle G}$  such that ${\displaystyle N\subseteq K\subseteq G}$ .
5. If ${\displaystyle K}$  is a normal subgroup of ${\displaystyle G}$  such that ${\displaystyle N\subseteq K\subseteq G}$ , then the quotient group ${\displaystyle (G/N)/(K/N)}$  is isomorphic to ${\displaystyle G/K}$ .

#### Theorem D

The correspondence theorem, aka the lattice theorem, is sometimes called the third or fourth isomorphism theorem.

The Zassenhaus lemma is sometimes called the "fourth isomorphism theorem", too.[citation needed]

### Discussion

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 ${\displaystyle f:G\rightarrow H}$ . 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 ${\displaystyle \iota \circ \pi }$ , 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 ${\displaystyle \ker f}$  and a monomorphism ${\displaystyle \kappa :\ker f\rightarrow G}$  (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 ${\displaystyle \ker f}$  to ${\displaystyle H}$  and ${\displaystyle G/\ker f}$ .

If the sequence is right split (i. e., there is a morphism σ that maps ${\displaystyle G/\ker f}$  to a π-preimage of itself), then G is the semidirect product of the normal subgroup ${\displaystyle \operatorname {im} \kappa }$  and the subgroup ${\displaystyle \operatorname {im} \sigma }$ . If it is left split (i. e., there exists some ${\displaystyle \rho :G\rightarrow \ker f}$  such that ${\displaystyle \rho \circ \kappa =\operatorname {id} _{\ker f}}$ ), then it must also be right split, and ${\displaystyle \operatorname {im} \kappa \times \operatorname {im} \sigma }$  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 ${\displaystyle \operatorname {im} \kappa \oplus \operatorname {im} \sigma }$ . In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence ${\displaystyle 0\rightarrow G/\ker f\rightarrow H\rightarrow \operatorname {coker} f\rightarrow 0}$ .

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.

## Modules

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 ${\displaystyle \mathbb {Z} }$ ) 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

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).

### Theorem B

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

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

1. If ${\displaystyle S}$  is a submodule of ${\displaystyle M}$  such that ${\displaystyle T\subseteq S\subseteq M}$ , then ${\displaystyle S/T}$  is a submodule of ${\displaystyle M/T}$ .
2. Every submodule of ${\displaystyle M/T}$  is of the form ${\displaystyle S/T}$ , for some submodule ${\displaystyle S}$  of ${\displaystyle M}$  such that ${\displaystyle T\subseteq S\subseteq M}$ .
3. If ${\displaystyle S}$  is a submodule of ${\displaystyle M}$  such that ${\displaystyle T\subseteq S\subseteq M}$ , then the quotient module ${\displaystyle (M/T)/(S/T)}$  is isomorphic to ${\displaystyle M/S}$ .

### Theorem D

Let ${\displaystyle M}$  be a module, ${\displaystyle N}$  a submodule of ${\displaystyle M}$ . There is a bijection between the submodules of ${\displaystyle M}$  which contain ${\displaystyle N}$  and the submodules of ${\displaystyle M/N}$ . The correspondence is given by ${\displaystyle A\leftrightarrow A/N}$  for all ${\displaystyle A\supseteq N}$ . This correspondence commutes with the processes of taking sums and intersections (i.e., is a lattice isomorphism between the lattice of submodules of ${\displaystyle M/N}$  and the lattice of submodules of ${\displaystyle M}$  which contain ${\displaystyle N}$ ).[12]

## General

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

A congruence on an algebra ${\displaystyle A}$  is an equivalence relation ${\displaystyle \Phi \subseteq A\times A}$  which forms a subalgebra of ${\displaystyle A\times A}$  considered as an algebra with componentwise operations. One can make the set of equivalence classes ${\displaystyle A/\Phi }$  into an algebra of the same type by defining the operations via representatives; this will be well-defined since ${\displaystyle \Phi }$  is a subalgebra of ${\displaystyle A\times A}$ . The resulting structure is the quotient algebra.

### Theorem A

Let ${\displaystyle f:A\rightarrow B}$  be an algebra homomorphism. Then the image of ${\displaystyle f}$  is a subalgebra of ${\displaystyle B}$ , the relation given by ${\displaystyle \Phi :f(x)=f(y)}$  (i.e. the kernel of f) is a congruence on ${\displaystyle A}$ , and the algebras ${\displaystyle \ A/\Phi \ }$  and ${\displaystyle \operatorname {im} f}$  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.)

### Theorem B

Given an algebra ${\displaystyle A}$ , a subalgebra ${\displaystyle B}$  of ${\displaystyle A}$ , and a congruence ${\displaystyle \Phi }$  on ${\displaystyle A}$ , let ${\displaystyle \Phi _{B}=\Phi \cap (B\times B)}$  be the trace of ${\displaystyle \Phi }$  in ${\displaystyle B}$  and ${\displaystyle [B]^{\Phi }=\{K\in A/\Phi :K\cap B\neq \emptyset \}}$  the collection of equivalence classes that intersect ${\displaystyle B}$ . Then

1. ${\displaystyle \Phi _{B}}$  is a congruence on ${\displaystyle B}$ ,
2. ${\displaystyle \ [B]^{\Phi }}$  is a subalgebra of ${\displaystyle A/\Phi }$ , and
3. the algebra ${\displaystyle [B]^{\Phi }}$  is isomorphic to the algebra ${\displaystyle B/\Phi _{B}}$ .

### Theorem C

Let ${\displaystyle A}$  be an algebra and ${\displaystyle \Phi ,\Psi }$  two congruence relations on ${\displaystyle A}$  such that ${\displaystyle \Psi \subseteq \Phi }$ . Then ${\displaystyle \Phi /\Psi =\{([a']_{\Psi },[a'']_{\Psi }):(a',a'')\in \Phi \}=[\ ]_{\Psi }\circ \Phi \circ [\ ]_{\Psi }^{-1}}$  is a congruence on ${\displaystyle A/\Psi }$ , and ${\displaystyle A/\Phi }$  is isomorphic to ${\displaystyle (A/\Psi )/(\Phi /\Psi )}$ .

### Theorem D

Let ${\displaystyle A}$  be an algebra and denote ${\displaystyle \operatorname {Con} A}$  the set of all congruences on ${\displaystyle A}$ . The set ${\displaystyle \operatorname {Con} A}$  is a complete lattice ordered by inclusion.[13] If ${\displaystyle \Phi \in \operatorname {Con} A}$  is a congruence and we denote by ${\displaystyle \left[\Phi ,A\times A\right]\subseteq \operatorname {Con} A}$  the set of all congruences that contain ${\displaystyle \Phi }$  (i.e. ${\displaystyle \left[\Phi ,A\times A\right]}$  is a principal filter in ${\displaystyle \operatorname {Con} A}$ , moreover it is a sublattice), then the map ${\displaystyle \alpha :\left[\Phi ,A\times A\right]\to \operatorname {Con} (A/\Phi ),\Psi \mapsto \Psi /\Phi }$  is a lattice isomorphism.[14][15]

## Note

1. ^ Jacobson (2009), sec 1.10
2. ^ van der Waerden, Algebra (1994).
3. ^ Durbin (2009), sec. 54
4. ^ [the names are] essentially the same as [van der Waerden 1994][3]
5. ^ Knapp (2016), sec IV 2
6. ^ Grillet (2007), sec. I 5
7. ^ Rotman (2003), sec. 2.6
8. ^ a b Milne (2013), Chap. 1, sec. Theorems concerning homomorphisms
9. ^ Scott (1964), secs 2.2 and 2.3
10. ^ I. Martin Isaacs (1994). Algebra: A Graduate Course. American Mathematical Soc. p. 33. ISBN 978-0-8218-4799-2.
11. ^ Paul Moritz Cohn (2000). Classic Algebra. Wiley. p. 245. ISBN 978-0-471-87731-8.
12. ^ Dummit and Foote (2004), p. 349
13. ^ Stanley and Sankappanavar (2012), p. 37
14. ^ Stanley and Sankappanavar (2012), p. 49
15. ^ 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

• 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, 1 (2nd ed.), Dover, ISBN 9780486471891
• Paul M. Cohn, Universal algebra, Chapter II.3 p. 57
• Milne, James S. (2013), Group Theory, 3.13
• van der Waerden, B. I. (1994), Algebra, 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.
• W. R. Scott (1964), Group Theory, Prentice Hall
• John R. Durbin (2009). Modern Algebra: An Introduction (6 ed.). Wiley. ISBN 978-0-470-38443-5.
• Anthony W. Knapp (2016), Basic Algebra (Digital second edition ed.)CS1 maint: extra text (link)
• Pierre Antoine Grillet (2007), Abstract Algebra (2 ed.), Springer
• Joseph J. Rotman (2003), Advanced Modern Algebra (2 ed.), Prentice Hall, ISBN 0130878685